 1. Introduction
 2. Terminology
 3. Quotient graphs
 4. Adjacency depth and model nets
 5. Linear graph knots
 6. Isotopy equivalence
 7. Group methods and maximal symmetry isotopes
 8. Classifying multicomponent entangled nets
 9. Classifying lattice nets
 10. Double lattice nets and further directions
 Supporting information
 References
 1. Introduction
 2. Terminology
 3. Quotient graphs
 4. Adjacency depth and model nets
 5. Linear graph knots
 6. Isotopy equivalence
 7. Group methods and maximal symmetry isotopes
 8. Classifying multicomponent entangled nets
 9. Classifying lattice nets
 10. Double lattice nets and further directions
 Supporting information
 References
lead articles
Isotopy classes for 3periodic net embeddings
^{a}Department of Mathematics and Statistics, Lancaster University, Bailrigg, Lancaster LA1 1SQ, United Kingdom, ^{b}Theoretische Chemie, Technische Universität Dresden, D01062 Dresden, Germany, ^{c}Dipartimento di Chimica, Università degli Studi di Milano, Milano 20133, Italy, and ^{d}Samara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Samara 443100, Russian Federation
^{*}Correspondence email: s.power@lancaster.ac.uk, baburinssu@gmail.com
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, nfold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3torus [0,1)^{3} is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.
type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth1 embedded nets with a singlevertex quotient graph are enumerated. Additionally, a classification is given of embeddings ofKeywords: periodic nets; embedded nets; coordination polymers; isotopy types; crystallographic frameworks.
1. Introduction
Entangled and interpenetrating coordination polymers have been investigated intensively by chemists in recent decades. Their classification and analysis in terms of symmetry, geometry and topological connectivity is an ongoing research direction (Batten & Robson, 1998; Carlucci et al., 2003, 2014; Blatov et al., 2004; Alexandrov et al., 2011). These investigations also draw on mathematical methodologies concerned with periodic graphs, group actions and classification (DelgadoFriedrichs, 2005; Koch et al., 2006; Schulte, 2014; Bonneau & O'Keeffe, 2015; Baburin, 2016). On the other hand, it seems that there have been few investigations to date on the dynamical aspects of entangled periodic structures with regard to deformations avoiding edge collisions, or with regard to excitation modes and flexibility in the presence of additional constraints (Guest et al., 2014). In what follows we take some first steps in this direction and along the way obtain some systematic classifications of basic families.
A proper linear 3periodic net N = (N,S) is a periodic bondnode structure in 3D with a set N of distinct nodes and a set S of noncolliding line segment bonds. The underlying structure graph G = G(N) is also known as the topology of N (cf. DelgadoFriedrichs et al., 2005). Thus, the net N is an embedded net for a topology G, it is translationally periodic with respect to each basis vector of some basis for the ambient space, the nodes are distinct points, and the bonds of N are noncolliding straightline segments between nodes. We also define the companion structure of a crystallographic barjoint framework C. In this case the bonds are of fixed lengths which must be conserved in any continuous motion. Additionally a 3periodic graph (G, T) is a pair in which a countable graph G carries a specific periodic structure T.
Formal definitions of the periodic entities C,N and G are given in Definitions 2.1, 2.4 and 2.5. In crystallographic terminology it is usual in such definitions to require connectedness (DelgadoFriedrichs & O'Keeffe, 2003). However, we find it convenient in these definitions to extend the usage to cover disconnected periodic structures.
Subclasses of linear dperiodic nets N in are defined in terms of the diversity of their connected components and we indicate the connections between these class divisions and those used for entangled coordination polymers. In particular, we define the dimension type, which gives a list of the periodic ranks of connected subcomponents, and the type which concerns the congruence properties between these components.
Fundamental to the structure of an embedded periodic net are its labelled quotient graphs which are finite edgelabelled graphs determined by periodicity bases. In particular the infinite structure graph G(N) is determined by any labelled quotient graph of this kind, and the (unique) quotient graph QG(N) is the graph of the labelled quotient graph of a primitive periodicity basis. These constructs for N provide useful discriminating features for embedded nets even if they are insensitive to entanglement and catenation.
Our main concern is the entangled nature of linear periodic nets in 3space which have more than one connected component; however we also consider the selfentanglement of connected structures. Specifically, we approach the classification of linear periodic nets in terms of a formal notion of periodic isotopy equivalence, as given in Definition 6.1. This asserts that two embedded periodic nets in are periodically isotopic if there is a continuous path of noncrossing embedded periodic nets between them which is associated with a continuous path of periodicity bases. In this way we formalize an appropriate variant of the notion of ambient isotopy which is familiar in the theory of knots and links.
As a tool for understanding periodic isotopy we define linear graph knots on the flat 3torus and their isotopy equivalence classes. Such a graph knot is a spatial graph in the 3torus which is a geometric realization (embedding) of the labelled quotient graph of a linear periodic net arising from a choice of righthanded periodicity basis for N. We prove a natural finiteness theorem (Theorem 6.4) showing that there are finitely many periodic isotopy types of linear graph knots with a given labelled quotient graph. This in turn implies that there are finitely many periodic isotopy types of linear 3periodic nets with a given labelled quotient graph.
Our discussions and results are structured as follows. Sections 2 to 6 cover terminology, illustrative examples and general underlying theory. In Section 7 we give group theory methods, while in Sections 8, 9 and 10 we give a range of results, determining periodic isotopy classes and topologies for various families of embedded nets.
More specifically, in Section 2 we give comprehensive terminology, ab initio, and give the connections with terms used for coordination polymers and with the net notations of both the Reticular Chemistry Structural Resource (RCSR) (O'Keeffe et al., 2008) and ToposPro (Blatov et al., 2014). In the key Section 3 we discuss labelled and unlabelled quotient graphs. The example considered in detail in Section 3.1 illustrates terminology and motivates the introduction of model nets for the analysis of periodic isotopy types (periodic isotopes). In Section 4 we define primitive periodicity bases and introduce a measure of adjacency depth for an embedded net. In Section 5, as preparation for the discussion of periodic isotopy for embedded nets, we define linear graph knots on the flat 3torus as spatial graphs with (generalized) line segment edges. In Section 6 we discuss various isotopy equivalences for graph knots. Also we define periodic isotopy equivalence for embedded nets and prove that it is an equivalence relation and that there are finitely many periodic isotopes with a common labelled quotient graph. In the group methods of Section 7 we give the group–supergroup construction of entangled nets (Baburin, 2016), the definition of maximal symmetry periodic isotopes, and the role of Burnside's lemma in counting periodic isotopes. In Section 8 we determine periodic isotopy classes and also restricted periodic isotopy classes for various multicomponent shift homogeneous embeddings of nfold pcu. Such multicomponent embedded nets are related to the interpenetrated structures with translationally equivalent components which are abundant in coordination polymers. For generic embeddings we give proofs based on Burnside's lemma for counting orbits of spatially equivalent embeddings, while for maximal symmetry embeddings for npcu we use computations based on group–supergroup methods. In Section 9 we give a detailed determination of the 19 topologies and periodic isotopy classes of connected linear 3periodic nets with a singlevertex quotient graph and adjacency depth 1 (Table 3). In the final section we indicate further research directions.
2. Terminology
In any investigation with crossdisciplinary intentions, in our case between chemistry (reticular chemistry and coordination polymers) and mathematics (isotopy types and periodic frameworks), it is important to be clear of the meaning of terms. Accordingly we begin by defining all terminology from scratch.
The structure graph G = (V, E) of a finite or countably infinite barjoint framework G is given a priori since, formally, a barjoint framework G in is a pair (G, p) consisting of a simple graph G, the structure graph, together with a placement map . The joints of G are the points p(v) and the bars of G are the (unordered) joint pairs p(v)p(w) associated with the edges vw in E. It is often assumed that p(v) ≠ p(w) for the edges vw and hence the bars may also be considered to be the associated nondegenerate line segments [p(v), p(w)].
A dperiodic barjoint framework in is a barjoint framework G = (G,p) in whose periodicity is determined by two sets F_{v}, F_{e} of noncoincident joints p(v) and bars p(u)p(w), respectively, together with a set of basis vectors for translational periodicity, say . The requirement is that the associated translates of the set F_{v} and the set F_{e} are, respectively, disjoint subsets of the set of joints and the set of bars whose unions are the sets of all joints and bars. In particular p is an injective map.
The pair of sets (F_{v}, F_{e}) is a building block, or repeating unit, for G. We refer to this pair of sets also as a motif for G for the basis and note that G is determined uniquely by any pair of periodicity basis and motif. In fact we shall only be concerned with finite motifs. Also, for 1 ≤ d′ < d we similarly define a d′periodic barjoint framework in as one that is generated by a finite motif and a linearly independent set of d′ vectors for translational periodicity.
Definition 2.1
A crystallographic barjoint framework C in , or crystal framework, is a dperiodic barjoint framework in with finitely many translation classes for joints and bars.
Viewing C as a barjoint framework, rather than as a geometric dperiodic net, is a conceptual prelude to the consideration of dynamical issues of flexibility and rigidity (Power, 2014a), one in which we may bring to bear geometric and combinatorial rigidity theory. Note however that we have not required a crystal framework to be connected.
In the case of a 3D crystal framework C, particularly an entangled one of material origin, it is natural to require that the line segments [p(v), p(w)], for vw ∈ E, are essentially disjoint in the sense that they intersect at most at a common endpoint p(x) for some x ∈ V. We generally adopt this noncrossing assumption and say that C is a proper crystal framework in this case. Thus a proper crystal framework C determines a closed set, denoted C, formed by the union of the (nondegenerate) line segments [p(v), p(w)], for vw ∈ E. We call this closed set the body of C. By our assumptions one may recover C from its body and the positions of the joints.
The connected components of a crystal framework may have a lower rank (or dimension) of periodicity. Accordingly we make the following definition.
Definition 2.2
A d′periodic crystal framework, with rank 1 ≤ d′ < d, is a d′periodic barjoint framework in , with d′ linearly independent period vectors and finitely many translation classes for joints and bars.
framework, orFor completeness we define a 0periodic barjoint framework in to be a finite barjoint framework in . Thus every connected component of a crystal framework in is either itself a crystal framework in or is a d′ ≤ d, or is a finite framework. Note that a subperiodic subframework exists for C if and only if C has infinitely many connected components, that is, if and only if the body of C has infinitely many topologically connected components.
framework with rank 1 ≤In view of the finiteness requirement for the d′periodic translation classes, a framework in has a joint set consisting of finitely many translates of a of rank d′. In general connected d′periodic subperiodic frameworks may differ in the nature of their affine span, or spatial dimension, which may take any integral value between d′ and d. Formally, the spatial dimension is the dimension of the linear span of all the socalled bar vectors p(w) − p(v) associated with the bars p(v)p(w) of the framework. Once again we define a subperiodic framework to be proper if there are no intersections of edges.
The various definitions above, and also the following definition of dimension type, transpose immediately to the simpler category of linear periodic nets N, as defined in the next section.
We now introduce the general terminology which is specific to 3D space. Also we indicate how later this formulation of dimension type aligns with the terminology used by chemists for entangled periodic nets.
Definition 2.3
A periodic or subperiodic framework C in 3D space has dimension type , where d′ is the periodicity rank of C and where d_{1}, …, d_{s} is the decreasing list of periodicity ranks of the connected components. (In the symbol, a number representing the periodicity rank of a component is listed only once even if it occurs as the rank of several components.)
In particular there are 15 dimension types d for periodicity rank3 crystallographic frameworks, or for linear 3periodic nets, namely
2.1. Categories of periodic structures
Consider the following frequently used terms for periodic structures, arranged with an increasing mathematical flavour: crystal, crystal framework, linear periodic net, periodic graph, topological crystal.
We have defined proper crystal frameworks in with essentially disjoint bars and these may be viewed as forming an `upper category' of periodic objects for which there is interest in barlengthpreserving dynamics. If we disregard bar lengths, but not geometry, then we are in the companion category of positions, or line drawings, or embeddings of dperiodic nets in . Such embeddings are of interest in reticular chemistry and in this connection we may define a linear dperiodic net in to be a pair (N, S), consisting of a set N of nodes and a set S of line segments, where these sets correspond to the joints and bars of a proper dperiodic crystal framework. A standalone definition is the following.
Definition 2.4
A (proper) linear dperiodic net in is a pair N = (N,S) where (i) S, the set of edges (or bonds) of N, is a countable set of essentially disjoint line segments [p, q], with p ≠ q, (ii) N, the set of vertices (or nodes) of N, is the set of endpoints of members of S, (iii) there is a basis of vectors for such that the sets N and S are invariant under the translation group T for this basis, (iv) the sets N and S partition into finitely many Torbits.
Thus a linear periodic net can be thought of as a proper linear embedding of the structure graph of a crystal framework, the relevant crystal frameworks being those with no isolated joints of degree 0. Note that a linear periodic net is not required to be connected.
A linear dperiodic net is referred to in reticular chemistry as an embedding of a `dperiodic net'. This is because the term dperiodic net has been appropriated for the underlying structure graph of a linear periodic net. See DelgadoFriedrichs & O'Keeffe (2003), for example. This reference, to a more fundamental category on which to build, so to speak, then allows one to talk of a dperiodic net having an embedding with, perhaps, certain symmetry attributes. It follows then, tautologically, that a dperiodic net is a graph with certain periodicity properties and we formally specify this in Definition 2.5.
The next definition is a slight variant of the definition given by DelgadoFriedrichs (2005), in that we also require edge orbits to be finite in number.
Definition 2.5
(i) A periodic graph is a pair (G, T), where G is a countably infinite simple (abstract) graph and T is a free abelian of Aut(G) which acts on G freely and is such that the sets of vertex orbits and edge orbits are finite. The group T is called a translation group for G and its rank is called the dimension of (G, T). (ii) A dperiodic graph or a dperiodic net is a periodic graph of dimension d. (iii) The translation group T and the periodic graph (G, T) are maximal if no periodic structure (G, T′) exists with T′ a proper of T.
The T [or the pair (G, T)] is referred to as a periodic structure on G. Some care is necessary with assertions such as `N is an embedding of a 3periodic net G'. This has two interpretations according to whether G comes with a given periodic structure T which is to be represented faithfully in the embedding as a translation group or, on the other hand, whether the embedding respects some periodic structure T′ in Aut(G).
Finally we remark that there is another category of nets which is relevant to more mathematical considerations of entanglement, namely stringnode nets in the sense of Power & Schulze (2018). In the discrete case these have a similar definition to linear periodic nets but the edges may be continuous paths rather than line segments.
2.2. Maximal symmetry, the RCSR and selfentanglement
Let N be a linear dperiodic net. Then there is a natural injective inclusion map
from the usual symmetry group (space group) of N to the group of its structure graph G(N).
Definition 2.6
Let N be a linear dperiodic net. (i) The graphical symmetry group of N is the group Aut(G(N)). This is also called the maximal symmetry group of N. (ii) A maximal symmetry embedding of G(N) is an embedded net M for which G(M) = G(N) and the map ι_{M} is a group isomorphism.
A key result of the work of DelgadoFriedrichs (2003, 2005) shows that many connected 3periodic graphs have unique maximal symmetry placements, possibly with edge crossings. These placements arise for a socalled stable net by means of a minimumenergy placement, associated with a fixed lattice of orbits of a single node, followed by a renormalization by the of the structure graph. In fact a stable net is defined as one for which the minimumenergy placement has no node collisions. See also DelgadoFriedrichs & O'Keeffe (2003), Sunada (2013). While maximumsymmetry positions for connected stable nets are unique, up to spatial congruence and rescaling, edge crossings may occur for simply defined nets because, roughly speaking, the local edge density is too high. It becomes an interesting issue then to define and determine the finitely many classes of maximumsymmetry proper placements and this is true also for multicomponent nets. See Section 7.1.
The RCSR (O'Keeffe et al., 2008) is a convenient online database which, in part, defines a set of around 3000 topologies G together with an indication of their maximal symmetry embedded nets. The graphs G are denoted in bold notation, such as pcu and dia, in what is now standard nomenclature. We shall make use of this and denote the maximal symmetry embedding of a connected topology abc as N_{abc}. This determines N_{abc} as a subset of up to a scaling factor and spatial congruence. ToposPro (Blatov et al., 2014) is a more sophisticated program package, suitable for multipurpose crystallochemical analysis and has a more extensive periodic net database. In particular it provides labelled quotient graphs for 3periodic nets.
Both these databases give coordination density data which can be useful for discriminating the structure graphs of embedded nets.
In Section 6 we formalize the periodic isotopy equivalence of pairs of embedded nets. One of our motivations is to identify and classify connected embedded nets which are not periodically isotopic to their maximal symmetry embedded net. We refer to such an embedded net as a selfentangled embedded net.
2.3. Derived periodic nets
We remark that the geometry and structural properties of a linear periodic net or framework can often be analysed in terms of derived nets or frameworks. These associated structures can arise through a number of operations and we now indicate some of these.
(i) The periodic substitution of a (usually connected) finite unit with a new finite unit (possibly even a single node) while maintaining incidence properties. This move is common in reticular chemistry for the creation of `underlying nets' (Alexandrov et al., 2011; O'Keeffe & Yaghi, 2012).
(ii) A more sophisticated operation which has been used for the classification of coordination polymers replaces each minimal ring of edges by a node (barycentrically placed) and adds an edge between a pair of such nodes if their minimal rings are entangled. In this way one arrives at the Hopf ring net (HRN) of an embedded net N (Alexandrov et al., 2012, 2017). This is usually well defined as a (possibly improper) linear 3periodic net and it has proven to be an effective discriminator in the taxonomic analysis of crystals and coordination polymer databases.
(iii) There are various conventions in which notational augmentation is used (O'Keeffe et al., 2008) to indicate the derivation of an embedded net or its relationship with a parent net. In the RCSR listing, for example, the notation pcuc4 indicates the topology made up of four disjoint copies of pcu (O'Keeffe et al., 2008). In Tables 1, 2 we use a notation for model embedded nets, such as M_{pcu}^{ff},…, which is indicative of a hierarchical construction.


(iv) On the mathematical side, in the rigidity theory of periodic barjoint frameworks C there are natural periodic graph operations and associated geometric moves, such as periodic edge contractions, which lead to inductive schemes in proofs. In particular periodic Henneberg moves, which conserve the average degree count, feature in the rigidity and flexibility theory of such frameworks (Nixon & Ross, 2015).
2.4. Types of entanglement and type
Let us return to descriptive aspects of disconnected linear 3periodic nets N in .
We first note the following scheme of Carlucci et al. (2014) which has been used in the classification of observed entangled coordinated polymers. Such a coordination polymer, P say, is also a proper linear dperiodic net N in , and this is either of full rank d = 3, or is of subperiodicity rank 1 ≤ d < 3, or is a finite net (which we shall say has rank 0). Let P be a dperiodic coordination polymer in . Then P is said to be (i) in the interpenetration class if all connected components of N are also dperiodic, (ii) in the polycatenation class otherwise.
Thus P is in the interpenetration class if and only if the dimension type of its net is {3; 3}, {2; 2}, {1; 1} or {0; 0}.
The entangled coordination polymers in the interpenetration class may be further divided as subclasses of nfold type, according to the number n of components, where, necessarily, n is finite.
The linear 3periodic nets in the polycatenation class have some components which are subperiodic and in particular they have countably many components. When all the components are 2periodic, that is, when N has dimension type {3; 2}, then N is either of parallel type or inclined type. Parallel type is characterized by the common coplanarity of the periodicity vectors of the components, whereas N is of inclined type if there exist two components which are not parallel in this manner. The diversity here may be neatly quantified by the number, ν_{2} say, of planes through the origin that are determined by the (pairs of) periodicity vectors of the components.
Similarly, the disconnected linear 3periodic nets of dimension type {3; 1} can be viewed as being of parallel type, with all the 1periodic components going in the same direction, or, if not, as inclined type. In fact there is a natural further division of the nonparallel (inclined) types for the nets of dimension type {3; 1} according to whether the periodicity vectors for the components are coplanar or not. We could describe such nets as being of coplanar inclined type and triple inclined type, respectively. The diversity here may also be neatly quantified by the number, ν_{1} say, of lines through the origin that are determined by the periodicity vectors of the components.
The disconnected nets of parallel type are of particular interest for their mathematical and observed entanglement features, such as Borromean entanglement and woven or braided structures (Carlucci et al., 2014; Liu et al., 2018).
The foregoing terminology is concerned with the periodic and subperiodic nature of the components of a net without regard to further comparisons between them. On the other hand, the following terms identify subclasses according to the possible congruence between the connected components.
N is of homogeneous type if its components are pairwise congruent. Here the implementing congruences are not assumed to belong to the space group.
N is nheterogeneous, with n > 1, if there are exactly n congruence classes of connected components.
Thus every 3periodic linear net N in is either of homogeneous type or nheterogeneous for some n = 2, 3, ….
The homogeneous linear 3periodic nets split into two natural subclasses.
N is of shifthomogeneous type if all components are pairwise translationally equivalent (shift equivalent). Otherwise, when N contains at least one pair of components which are not shift equivalent then we say that the homogeneous net N is of rotation type.
Finally we take account of the N to specify a very strong form of each of the two homogeneous types contains a further subtype according to whether N is also of transitive type or not, where N is component transitive (or is of transitive type) if the of N acts transitively on components.
ofSuch component transitive periodic nets have been considered in detail by Baburin (2016) with regard to their construction through group–supergroup methods.
Note that a homogeneous linear 3periodic net in which is not connected falls into exactly one of 16 possible dimension–homogeneity types according to the four possible types of .
and the four possible dimension types , namely or {3; 0}. For a full list of correspondences see Fig. 12.5. Catenation and Borromean entanglement
To the dimension–homogeneity type division of multicomponent embedded nets one may consider further subclasses which are associated with entanglement features between the components. Indeed, our main consideration in what follows is a formalization of such entanglement in terms of linear graph knots. We note here some natural entanglement invariants of Borromean type. In fact the embedded nets of dimension type {3; 2} have been rather thoroughly identified by Carlucci et al. (2014), Alexandrov et al. (2017) where it is shown that subdimensional 2periodic components can be catenated or woven in diverse ways.
To partly quantify this one may define the following entanglement indices. Let N be such a parallel type embedded net, with dimension type {3; 2}, and let S be a finite set of components. Then a separating isotopy of S is a continuous deformation of S to a position which properly lies on both sides of the complement of a plane in . If there is no separating isotopy for a pair S = {N_{i},N_{j}} of components of N then we say that they are entangled components, or are properly entangled. This partial definition can be made rigorous by means of a formal definition of periodic isotopy. We may then define the component entanglement degree of a component N_{i} of N to be the maximum number, δ(N_{i}) say, of components N_{j} which can form an entangled pair with N_{i}. Also the component entanglement degree of N itself may be defined to be the maximum such value.
Likewise, one can define the entanglement degrees of components for embedded nets of dimension type {3; 1} and for the subdimensional nets of dimension type {2; 2} (woven layers) and dimension type {1; 1} (braids). More formally, we may say that N has Borromean entanglement if there is a set of n ≥ 3 connected components which admit no separating periodic isotopy while, on the other hand, every pair in this set admits a separating periodic isotopy. In a similar way one can formalize the notion of Brunnian catenation (Liang & Mislow, 1994) for a multicomponent embedded net.
2.6. When topologies are different
Two standard graph isomorphism invariants used by crystallographers are the point symbol and the coordination sequence.
In a vertex transitive countable graph G the point symbol (PS), which appears as 4^{24}6^{4} for bcu for example, indicates the multiplicities (24 and 4) of the cycle lengths (4 and 6) for a set of minimal cycles which contain a pair of edges incident to a given vertex. If the valency (or coordination) of G is r then there are r(r − 1)/2 such pairs and so the multiplicity indices sum to r(r − 1)/2. For G nontransitive on vertices the point symbol is a list of individual point symbols for the vertex classes (Blatov et al., 2010).
The coordination sequence (CS) of a vertex transitive countable graph G is usually given partially as a finite list of integers associated with a vertex v, say n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, where n_{k} is the number of vertices w ≠ v for which there is an edge path from v to w of length k but not of shorter length. For bcu this sequence is 8, 26, 56, 98, 152. Cumulative sums of the CS are known as topological densities, and the RCSR, for example, records the tenfold sum, td10.
Even the entire CS is not a complete invariant for the set of underlying graphs of embedded periodic nets. However, this counting invariant can be useful for discriminating nets whose local structures are very similar. A case in point is the pair 8T17 and 8T21 appearing in Table 3, which have partial coordination sequences 8, 32, 88 and 8, 32, 80, respectively.

3. Quotient graphs
We now define quotient graphs (QGs) and labelled quotient graphs (LQGs) associated with the periodic structure bases of a linear periodic net N. Although QGs and LQGs are not sensitive to entanglement they nevertheless offer a means of subcategorizing linear periodic nets. See, for example, the discussions by Eon (2011, 2016), Klee (2004), Klein (2012), Thimm (2004) and Section 4.3 below.
Let N = (N,S) be a linear 3periodic net with periodic structure basis . Then N is completely determined by and any associated building block motif (F_{v}, F_{e}). It is natural, especially in illustrating examples, to choose the set F_{e} of edges of N to be as connected as possible and to choose F_{v} to be a subset of the vertices of these edges. Let T denote the translation group associated with , so that T is the set of transformations
Each (undirected) line segment edge p(e) in F_{e} has the form [T_{k}p(v_{e}), T_{l}p(w_{e})], where p(v_{e}) and p(w_{e}) are the representatives in F_{v} for the endpoint nodes T_{k}p(v_{e}), T_{l}p(w_{e}) of the edge p(e). The labels k and l here may be viewed as the cell labels or translation labels associated with endpoints of p(e). [As before v_{e}, w_{e} indicate vertices of the underlying structure graph G(N).]
The LQG of the pair is a finite multigraph together with a directed labelling for each edge, where the labelling is by elements . The vertices correspond to (or are labelled by) the vertices v of the nodes p(v) in F_{v}, and the edges correspond to edges p(e) in F_{e}. The directedness is indicated by the ordered pair (v_{e}, w_{e}), or by v_{e}w_{e} (viewed as directedness `from v_{e} to w_{e}'). The label for this directed edge is then k − l where k, l are the translation labels as in the previous paragraph, and so the labelled directed edge is denoted (v_{e}w_{e}, k − l). There is no ambiguity since the directed labelled edge (v_{e}w_{e}, k − l) is considered to be the same directed edge as (w_{e}v_{e}, l − k). In particular the following definition of the depth of a labelled directed graph is well defined.
Definition 3.1
Let H = (H,λ) be any LQG. Then the depth of H is the maximum modulus of the coordinates of the edge labels.
The QG of the pair is the undirected graph G obtained from the LQG. If is a primitive periodicity basis, that is, one associated with a maximal lattice in N, then QG is independent of and is the usual quotient graph of N in which the vertices are labelled by the translation group orbits of the nodes. Primitive periodicity bases are discussed further in the next section. Moreover, we identify there the `preferred' primitive periodicity bases which have a `best fit' for N in the sense of minimizing the maximum size of the associated edge labels.
Definition 3.2
The QG(N) of a linear periodic net N in is the unlabelled multigraph graph of the LQG determined by a primitive periodicity basis.
Finally we remark on the homological terminology related to the edge labellings of a LQG. The homology group of the 3torus is isomorphic to . In this isomorphism the standard generators of may be viewed as corresponding to (homology classes of) three 1cycles which wind once around the 3torus (which we may parametrize naturally by the set [0, 1)^{3}) in the positive coordinate directions. Also, we may associate the standard ordered basis for with a periodicity basis for N. In this case the sum of the labels of a directed cycle of edges in the labelled quotient graph is equal to the homology class of the associated closed path in .
3.1. Embedded nets with a common LQG
We now consider the family of all linear 3periodic nets (proper embedded nets) which have a periodic structure basis determining a particular common LQG. This discussion illuminates some of the terminology set out so far and it also gives a prelude to discussions of periodic isotopy. Also it motivates the introduction of model nets and linear graphs knots on the 3torus.
Let H be the 6coordinated graph with two vertices v_{1}, v_{2}, two connecting edges between them and two loop edges on each vertex. (We say that a finite or countable graph is ncoordinated if every vertex has valency n.) Let (H, λ) be the LQG with labels (0, 0, 1), (1, 1, 1) for the loop edges for v_{1}, labels (0, 1, 0), (0, 0, 1) for the loop edges for v_{2}, and labels (0, 0, 0), (0, −1, −1) for the two directed edges from v_{1} to v_{2}. Let N be an embedded net with a general periodic structure basis such that LQG. (In particular N has adjacency depth 1, as defined in the next section.) Note that the four loop edges on v_{1} and v_{2} imply that N has two countable sets of 2D parallel subnets all of which are pairwise disjoint. These subnets are either parallel to the pair {a_{2}, a_{3}} or to the pair {a_{3}, a_{1} + a_{2} + a_{3}}. In particular if N^{′} ⊆ N is the embedded net which is the union of these 2D subnets then N^{′} is a derived net of N of dimension type {3; 2}. Also N^{′} is in the polycatenation class of inclined type (rather than parallel type). By means of a simple oriented affine equivalence (see Definition 4.3) the general pair with LQG(H, λ) is equivalent to a pair , having the same LQG, where is the standard righthanded orthonormal basis. We shall call the pair a model net.
By translation (another oriented affine transformation) we may assume that there is a node p_{1} of M at the origin which is associated with the vertex v_{1} of H. Let p_{2} be the unique node associated with v_{2} which lies in the [0, 1)^{3}. Now the pair is uniquely determined by p_{2} and we denote it simply as M(p_{2}). Fig. 2 illustrates the part of the linear periodic net M(p_{2}) which is visible in [0, 1)^{3}. In Section 5 we shall formalize diagrams such as Fig. 2 in terms of linear graph knots on the flat 3torus.
With this normalization the point p_{2} can be any point in [0, 1)^{3}, subject to the essential disjointness of edges, and we write O for this set of positions of p_{2}. Note that as p_{2} moves on a small closed circular path around the main diagonal its incident edges are determined and there will be five edge crossings with the diagonal. In fact the two vertical edges and the two horizontal edges which are incident to p_{2} contribute two crossings each, and the other edge incident to p_{2} contributes one crossing. These are the only edge crossings that occur as p_{2} `carries' its six edges of incidence during this motion. It follows from similar observations that O is the disjoint union of five pathwise connected sets.
In this way we see that a pair of nets M(p_{2}),M(p_{2}^{′}), with p_{2}, p_{2}′ in the same component set, are strictly periodically isotopic in the sense that there is a continuous path of linear periodic nets between them, each of which has the same periodic structure basis, namely . From this we may deduce that there are at most five periodic isotopy classes of embedded nets N which have the specific LQG (H, λ) for some periodic structure. Conceivably there could be fewer periodic isotopy classes since we have not contemplated isotopy paths of nets, with associated paths of periodicity bases, for which the LQG changes several times before returning to (H, λ).
Let us also note the following incidental facts about the nets M(p_{2}). They are 6coordinated periodic nets and so provide examples of critically coordinated barjoint frameworks, of interest in rigidity theory and the analysis of rigid unit modes. This is also true of course for all frameworks with the same underlying quotient graph.
4. Adjacency depth and model nets
We now define the adjacency depth of a linear 3periodic net N. This positive integer can serve as a useful taxonomic index and in Sections 9, 10 we determine, in the case of some small quotient graphs, the 3periodic graphs which possess an embedding as a (proper) linear 3periodic net with depth 1. These identifications also serve as a starting point for the determination of the periodic isotopy types of more general depth1 embedded nets.
We first review maximal periodicity lattices for embedded nets N and their primitive periodicity bases.
4.1. Primitive periodic structure
Let be a dperiodic net N. The associated translation group of isometries of N is a of the of N. We say that is a primitive, or a maximal periodicity basis, if there is no periodicity basis such that is a proper subset of .
basis for which consists of a periodicity basis for a linearWe focus on 3D and in order to distinguish mirrorrelated nets we generally consider righthanded periodicity bases of the embedded nets N.
The next wellknown lemma shows that different righthanded primitive bases are simply related by the matrix of an invertible transformation with integer entries and determinant 1. Let be the group of invertible d × d real matrices, viewed also as linear transformations of , and let be the of matrices with positive determinant. Also, let be the of elements with determinant 1, and the of with integer entries.
4.2. The adjacency depth ν(N) of a linear periodic net
While certain elementary linear periodic nets N have `natural' primitive periodicity bases, it follows from Lemma 4.1 that such a basis is not determined by N. It is natural then to seek a preferred basis which is a `good fit' in some sense. The next definition provides one such sense, namely that the should be one that minimizes the adjacency depth of the pair .
Definition 4.2
The adjacency depth of the pair , denoted , is the depth of the LQG, that is the maximum modulus of its edge labels. The adjacency depth, or depth, of N is the minimum value, ν(N), of the adjacency depths taken over all righthanded primitive periodicity bases .
Let N be a linear 3periodic net with periodicity basis . Consider the semiopen parallelepipeds (rhomboids)
These sets form a partition of , with P_{k} viewed as a with label k. Note that each cell P_{j} has 26 `neighbours', given by those cells P_{l} whose closures intersect the closure of P_{k}. (For diagonal neighbours this intersection is a single point.) Thus we have the equivalent geometric description that ν(N) = 1 if and only if there is a primitive periodicity basis such that the pair of end nodes of every edge lie in neighbouring cells of the cell partition, where here we also view each cell as a neighbour of itself.
It should not be surprising that for the connected embedded periodic nets of materials the adjacency depth is generally 1. Indeed, while the maximum symmetry embedding N_{elv} for the net elv has adjacency depth 2, it appears to us to be the only connected example in the current RCSR listing with . The periodic net elv gets its name from the fact that its minimal edge cycles have length 11. On the other hand, in Section 8 we shall see simple examples of multicomponent nets with adjacency depth equal to the number of connected components.
Definition 4.3
Let N_{i} = (N_{i},S_{i}), i = 1, 2, be linear 3periodic nets in . Then N_{1} and N_{2} are affinely equivalent [respectively, orientedly (or chirally) affinely equivalent] if there are translates of N_{1} and N_{2} which are conjugate by a matrix X in [respectively, ].
It follows from the definitions that if N_{1} and N_{2} are affinely equivalent then they have the same adjacency depth.
The next elementary lemma is a consequence of the fact that linear 3periodic nets are, by assumption, proper in the sense that their edges must be noncrossing (i.e. essentially disjoint).
Lemma 4.4
Let N be a linear 3periodic net with a depth1 LQG (H, λ). Then there are at most seven loop edges on each vertex of H and the multiplicity of edges between each pair of vertices is at most 8.
Proof
Let be a periodic structure basis such that . Without loss of generality we may assume that is an orthonormal basis. Let p_{1} be a node of N. Let p_{2}, …, p_{8} be the nodes T_{k}p_{1} where k ≠ (0, 0, 0) with coordinates equal to 0 or 1, and let p_{9}, …, p_{27} be the nodes T_{k}p_{1} for the remaining values of k with coordinates equal to 0, 1 or −1. Every line segment [p_{1}, p_{t}] with t ≥ 9 has a lattice translate which either coincides with or intersects, at midpoints, one of the line segments [p_{1}, p_{t}] with t < 9. Since N has no edge crossings it follows that there are at most seven translation classes for the edges associated with multiple loops of H at a vertex.
We may assume that p_{1} = (0, 0, 0). Let q_{1} be a node in (0, 1)^{3} in a distinct translation class. Since the depth is 1 it follows that the edges [q_{1}, p] in N with p a translate of p_{1}, correspond to the positions p = λ, where has coordinates taking the values −1, 0 or 1. The possible values of λ are also the labels in the quotient graph of N for the edges directed from the orbit vertex of p_{1} to the orbit vertex of q_{1}. There are thus 27 possibilities for the edges [q_{1}, p], and we denote the terminal nodes p by λ_{a}, λ_{b}, ….
Since N is a proper net, with no crossing edges, we have the constraint that k = (λ_{a} − λ_{b})/2 is not a lattice point for any pair λ_{a}, λ_{b}. For otherwise [q_{1} + k, λ_{b} + k] is an edge of N and its midpoint coincides with the midpoint of [q_{1}, λ_{a}]. It follows from the constraint that there are at most eight terminal nodes.
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The following proposition gives a necessary condition for a general 3periodic graph (G, T) to have an embedding as a proper linear 3periodic net. Moreover this condition is useful later for the computational determination of possible topologies for nets which have a quotient graph with one or two vertices.
We say that a LQG (H, λ) has the divisibility property, or is divisible, if for some pair of labelled edges (v_{1}v_{2}, k), (v_{1}v_{2}, l), with the same vertices, and possibly v_{1} = v_{2}, the vector k − l is divisible in the sense that it is equal to nt, with and n ≥ 2 an integer. If this does not hold then the three entries of k − l are coprime and (H, λ) is said to be indivisible.
Proposition 4.5
Let N be a (proper) linear 3periodic net in and let (H, λ) be a LQG associated with some periodic structure basis for N. Then (H, λ) is indivisible.
Proof
Let (v_{1}v_{2}, k), (v_{1}v_{2}, l) be two edges of (H, λ), with v_{1} ≠ v_{2}. Then N has the incident edges [(p(v_{1}), 0), (p(v_{2}), k)], [(p(v_{1}), 0), (p(v_{2}), l)] which, by the properness of N, are not collinear. Without loss of generality and to simplify notation assume that the periodicity basis defining the LQG is the standard orthonormal basis. Then these edges are [p(v_{1}), p(v_{2}) + k] and [p(v_{1}), p(v_{2}) + l]. Taking all translates of these two edges by integer multiples of t = k − l we obtain a 1periodic (zigzag) subnet, Z say, of N with period vector t = (t_{1}, t_{2}, t_{3}).
Suppose next that t is divisible with and n ≥ 2. Since Z+t^{′} does not coincide with Z there are crossing edges, a contradiction.
Consider now two loop edges (v_{1}v_{1}, k), (v_{1}v_{1}, l) and corresponding incident edges in N, say [p(v_{1}), p(v_{1}) + k] and [p(v_{1}), p(v_{1}) + l]. Taking all translates of these two edges by the integer combinations n_{1}k + n_{2}l, with , we obtain a 2periodic subnet, with period vectors {k, l}, which is an embedding of sql. The vector t = k − l is a diagonal vector for the parallelograms of this subnet and so, as before, t cannot be divisible.
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As a consequence of the proof we also see that an embedded net is improper if either of the following conditions fails to hold: (i) for pairs of loop edges in the LQG with the same vertex the two labels generate a maximal rank2
of the translation group, (ii) for pairs of nonloop edges the difference of the two labels generates a maximal rank1 subgroup.4.3. Model nets and LQGs
We first note that every abstract 3periodic graph (G, T) can be represented by a model net M in with standard periodicity basis , in the sense that G is isomorphic to the structure graph G(M) of M by an isomorphism which induces a representation of T as the translation group of M associated with . Formally, we define a model net to be such a pair but we generally take the basis choice as understood and use notation such as M,M(p,i),M(p_{2}) etc.
Let (G, T) be a 3periodic graph with periodic structure T and let H = G/T be the quotient graph (V(H), E(H)) determined by T. Identify the group T with the integer translation group of . This is achieved through the choice of a and this choice introduces an ordered triple of generators and coordinates for T. Any other such map, j say, has the form X °i where .
Label the vertices of G by pairs (v_{k}, g) where g ∈ T and v_{1}, …, v_{n} is a complete set of representatives for the Torbits of vertices. For the sake of economy we also label the vertices of H by v_{1}, …, v_{n}. Let p_{H} : V(H) → [0, 1)^{3} be any injective placement map. Then there is a unique injective placement map induced by p and i, with
Thus the maps p_{H}, i determine a (possibly improper) model embedded net for (G, T) which we denote as M(p,i). In particular if ((v_{k}, g), (v_{l}, h)) is an edge of G then this determines the line segment edge [p((v_{k}, g)), p((v_{l}, h))] of M(p,i). This net is possibly improper since some edges may intersect. Write H(i) for the LQG (H, λ) of M(p,i) with respect to . As the notation implies, this depends only on the choice of i which coordinatizes the group T.
With i fixed we can consider continuous paths of such placements, say p_{H}^{t},0 ≤ t ≤ 1, which in turn induce paths of model nets, t→M_{t} = M(p_{H}^{t}),0 ≤ t ≤ 1. (See also Section 3.1.) When there are no edge collisions, that is, when all the nets M_{t} in the path are proper, this provides a strict periodic isotopy between the the pairs and and their given periodic structure bases, . (Such isotopy is also formally defined in the remarks following Definition 6.1.)
Note that if H is a bouquet graph, that is, has a single vertex, then the strict periodic isotopy determined by t→ p_{H}^{t} between two model nets for H corresponds simply to a path of translations.
In the next proposition we consider 3periodic graphs as pairs (G, T), as in Definition 2.5 [and Definition 4.2 of Eon (2011)]. Moreover we have the following natural notion of isomorphism.
Definition 4.6
The pairs (G, T), (G′, T′) are isomorphic as 3periodic graphs if and only if there is a countable graph isomorphism G → G′ induced by a bijection γ : V → V′, together with a π : T → T′ such that γ(g(v)) = π(g)(γ(v)) for v ∈ V, g ∈ T.
It is in this sense that we may say that an isomorphism (G, T) → (G′, T′) of periodic graphs is a pair of isomorphisms (γ, π) which respects the periodic structure.
Note, for example, that the countable structure graph G = G(N_{pcu}) has periodic structure T (respectively, T′) determined by the periodicity basis (2, 0, 0), (0, 2, 0), (0, 0, 1) [respectively, (4, 0, 0), (0, 1, 0), (0, 0, 1)] for N_{pcu}. The periodic graphs (G, T) and (G, T′) fail to be isomorphic since they have different quotient graphs, which is a necessary condition for this.
Proposition 4.7
Let (G, T), (G′, T′) be 3periodic graphs (with given periodic structures) with LQGs H(i) = (H,λ), H(i^{′}) = (H^{′},λ^{′}) arising from group isomorphisms and . Then the following statements are equivalent.
(i) (G, T) and (G′, T′) are isomorphic as 3periodic graphs.
(ii) There is a graph isomorphism ϕ : H → H′ and with detX = 1 such that λ′(ϕ(e)) = X(λ(e)) for all directed edges e of H(i).
Proof
(ii) ⇒ (i). A typical edge e of H(i) is denoted by a triple [v_{e}, w_{e}, λ(e)] and a typical associated edge of G is
where g ∈ T and where we have written the group operation in T additively. Define γ : V(G) → V(G′) by γ(g(v)) = π(g)(ϕ(v)) where v ∈ V(H), g ∈ T and π is the T → T′ defined by π = (i′)^{−1} °X °i. Then γ is a bijection between the vertex sets of G and G′. Moreover, we note that since π°i^{−1} is equal to (i′)^{−1} °X the γinduced edge [γ(g(v_{e}), γ(g + i^{−1}(λ(e)))(w_{e})] is equal to
and so is an edge of G′, since X(λ(e)) is equal to λ′(ϕ(e)), the label of the edge ϕ(e) in H(i^{′}). Thus γ induces a graph isomorphism G → G′ and moreover the pair γ, π is an isomorphism of the periodic graphs, as required.
(i) ⇒ (ii). Consider an isomorphism from (G, T) to (G′, T′) given by the pair γ, π. Note that γ_{v} : V(G) → V(G′) maps Torbits to T′orbits, as does γ_{e} : E(G) → E(G′), and so γ defines a graph isomorphism ϕ from H = G/T to H′ = G′/T′. Also, the edge [v_{e}, i^{−1}(λ(e))(w_{e})] in G [associated with the edge e = (v_{e}, w_{e}, λ(e)) as before] maps to
where X is the matrix in with X = i′°π°i^{−1}. This implies that Xλ(e) must be the label for the associated edge ϕ(e) in H^{′}(i^{′}), and so (ii) holds.
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In the case when H = G/T and H′ = G/T′ are bouquet graphs one can say much more. Any graph isomorphism γ : G → G′ lifts to a linear isomorphism between the model nets M,M^{′} determined by any pair T, T′ of maximal periodic structures. See for example Proposition 3 of Kostousov (2007). It follows that for bouquet quotient graph nets we have the following stronger theorem.
Theorem 4.8
Let M(p,i) and M(p^{′},i^{′}) be model nets, with nodes on the integer lattice, for 3periodic graphs (G, T) and (G, T′) with bouquet quotient graphs. Then the following are equivalent. (i) G and G′ are isomorphic as countable graphs. (ii) M(p,i) and M(p^{′},i^{′}) are affinely equivalent by a matrix X in .
Definition 4.9
A (proper) linear 3periodic net N is a lattice net if its set of nodes is a lattice in .
Equivalently N is a lattice net if its quotient graph is a bouquet graph. One may also define a general lattice net in as a (not necessarily proper) embedded net whose quotient graph is a bouquet graph. Theorem 4.8 shows that lattice nets (even general ones) are classified up to affine equivalence by their topologies. In Theorem 9.5 we obtain a proof of this in the depth1 case through a casebycase analysis. Also we show that for the connected depth1 lattice nets there are 19 classes.
In principle Proposition 4.7 could be used as a basis for a computational classification of periodic nets with small quotient graphs with a depth1 labelling. However we note that there are more practical filtering methods such as those underlying Proposition 10.1 which determines the 117 connected topologies associated with certain depth1 nets which are supported on two parallel vertex lattices in a bipartite manner.
5. Linear graph knots
Let H be a multigraph, that is, a general finite graph, possibly with loops and with an arbitrary multiplicity of `parallel' edges between any pair of vertices. Then a graph knot in is a faithful geometric representation of H where the vertices v are represented as distinct points p(v) in and each edge e with vertices v, w is represented by a smooth path , with endpoints p(v), p(w). Such paths are required to be free from selfintersections and disjoint from each other, except possibly for coinciding endpoints. Thus a graph knot K is formally a triple , and we may also refer to this triple as a spatial graph or as a proper placement of H in . It is natural also to denote a graph knot K simply as a pair (N, S), where N is the set of vertices, or nodes, p(v) in , and S is the set of nonintersecting paths . We remark that spatial graphs feature in the mathematical theory of intrinsically linked connected graphs (Conway & Gordon, 1983; Kohara & Suzuki, 1992).
One can similarly define a graph knot K in any smooth manifold and of particular relevance is the Riemannian manifold known as a flat 3torus. This is essentially the topological 3torus identified naturally with the set [0, 1)^{3} and the topology, in the usual mathematical sense, is the natural one associated with continuity of the quotient map . Moreover we define a line segment in the flat torus to be the image of a line segment in under this quotient map. The curiosity here is that such a flat torus line segment may appear as the union of several line segment sets in [0, 1)^{3}.
We formally define a linear graph knot in the flat torus to be a triple , or a pair (N, S), where the vertices, or nodes, p(v) lie in [0, 1)^{3} and the paths, or edges, , are essentially disjoint flat torus line segments. Intuitively, this is simply a finite net in the flat 3torus with linear nonintersecting edges.
We now associate a linear graph knot K in [0, 1)^{3} with an embedded net N with a specified periodicity basis . Informally, this is done by replacing N by its affine normalization N^{′}, wherein is rescaled to the standard basis, and defining K as the intersection of the body N^{′} with [0, 1)^{3}. That is, one takes the simplest model net N^{′} for N and ignores everything outside the cube [0, 1)^{3}.
For the formal definition, let (F_{v}, F_{e}) be a motif for , where F_{v} ⊂ N (respectively, F_{e} ⊂ S) is a finite set of representatives for translation classes of nodes (respectively, edges) of N = (N,S), with respect to . Let be the natural quotient map associated with the ordered basis . This is a composition of the linear map for which maps to the standard righthanded basis, followed by the quotient map. Define p : F_{v} → [0, 1)^{3} to be the induced injection and to be the induced map from closed line segments to closed line segments of the flat torus [0, 1)^{3}.
Definition 5.1
Let H be the quotient graph for the pair . The triple is the linear graph knot of and is denoted as .
Since N is necessarily proper, with essentially disjoint edges, the placement has essentially disjoint edges and so is a linear graph knot.
Note that the linear graph knot determines uniquely the net N^{′} which in fact can be viewed as its covering net. It follows immediately that if then N and M are linear periodic nets which are orientedly affine equivalent.
We now give some simple examples together with perspective illustrations. Such illustrations are unique up to translations within the flat 3torus and so it is always possible to arrange that the nodes are interior to the open unit cube. In this case the 3D diagram reveals their valencies. On the other hand, as we saw in the partial body examples in Section 3.1 it can be natural to normalize and simplify the depiction by a translation which moves a node to the origin.
Example 5.2
The simplest proper linear 3periodic net is the primitive cubic net N = N_{pcu}. We may normalize this so that the node set is a translate of the set . The standard primitive periodic structure basis gives the graph knot , which we denote as K_{pcu} and which is illustrated in Fig. 3. The three `line segment' edges in the flat torus are here depicted by three pairs of line segments. The quotient graph of N_{pcu}, which is also the underlying graph of K_{pcu}, has one vertex and three loop edges. Note that if the node is translated to the origin then the depiction of the loop edges is given by three axial line segments.
By taking a union of n disjoint generic translates (within [0, 1)^{3}) of K_{pcu} one obtains the linear graph knot of an associated multicomponent linear net. In Theorem 8.2 we compute the number of periodic isotopy classes of such nets and the graph knot perspective is helpful for the proof of this.
Example 5.3
Fig. 4 shows linear graph knots (or finite linear nets) on the flat torus for the maximal symmetry nets N_{bcu} and N_{srs}. Each is determined by a natural primitive righthanded depth1 periodicity basis which, by the definition of , is normalized to . The quotient graphs for these examples are, respectively, the bouquet graph with four loop edges and the complete graph on four vertices. The periodic extensions of these graph knots give welldefined model nets, say M_{bcu} and M_{srs}, which are orientedly affinely equivalent to the maximal symmetry nets N_{bcu} and N_{srs}.
Example 5.4
The linear 3periodic net N_{dia} for the diamond crystal net (with maximal symmetry) has a periodic structure basis corresponding to three incident edges of a regular tetrahedron, and has a motif consisting of two vertices and four edges. The graph knot is obtained by (i) an oriented affine equivalence with a model net M_{dia} with standard orthonormal periodic structure basis, and (ii) the intersection of M_{dia} with [0, 1)^{3}. This graph knot has an underlying graph H(0, 4, 0) (in the notation of Section 10) with two vertices and four nonloop edges.
In Figs. 5, 6 we indicate four graph knots which define model nets each with underlying net (structure graph) equal to dia. In fact the graph knots K_{1}, K_{2} are rotationally linearly isotopic (see Definition 6.3). To see this consider a linear graph knot homotopy starting with K_{1} which is determined by a downward motion of the central vertex [at (1/2, 1/2, 1/2) say] of K_{2} through the floor of [0, 1)^{3}. The edge deformations are determined and, since the floor is equal to the roof, we can terminate the vertex motion at (1/2, 1/2, 1/2). Note that there are no edge crossings, so that the homotopy is in fact an isotopy. Moreover, examining the edges, one of which is reentrant, we see that the final linear graph knot is equal to the image of K_{1} under a halfturn rotation about the line through (1/2, 1/2, 1/2) and (0, 0, 1/2).
On the other hand K_{2} and K_{3} are linearly isotopic in terms of a motion of the vertex of K_{2} at the origin to the position of the lefthand vertex of K_{3}. It follows from this that the associated model nets M_{1},M_{2},M_{3} are strictly periodically isotopic, simply by taking the periodic extension of these isotopies to define periodic isotopies.
In contrast to this, observe first that the linear graph knot K_{4} is obtained from K_{3} by a continuous motion of the nodes p_{1} and p_{2} to their new positions in the 3torus. Such a motion defines a linear homotopy in the natural sense. The (uniquely) determined edges of the intermediate knots in this case inevitably cross at some point in the motion so these linear homotopies are not linear isotopies. The model net M_{4} for the knot K_{4} is in fact not periodically isotopic to the unique maximal symmetry embedding N_{dia}, and so is selfentangled. We show this in Example 6.7.
In the model nets of the examples above we have taken a primitive periodic structure basis with minimal adjacency depth. In view of this the represented edges between adjacent nodes in these examples have at most two diagramatic components, that is they reenter the cube at most once. In general the linear graph knot associated with a periodic structure basis of depth 1 has edges which can reenter at most three times.
Remark 5.5
We shall consider families of embedded nets up to oriented affine equivalence and up to periodic isotopy. In general there may exist enantiomorphic pairs, that is, mirror images N,N^{′} which are not equivalent. This is the case, for example, for embeddings of srs. However, such inequivalent pairs do not exist if the quotient graph is a single vertex (lattice nets) or a pair of vertices with no loop edges (double lattice nets with bipartite structure). This becomes evident in the latter case, for example, on considering an affine equivalence with a model net for which the point (1/2, 1/2, 1/2) is the midpoint of the two representative nodes in the This midpoint serves as a point of inversion for the model net (or, equivalently, its graph knot). The graph knots in Fig. 6 indicate such centred positions.
Remark 5.6
We have observed that for a linear 3periodic net the primitive righthanded periodicity bases are determined up to transformations by matrices in . Such matrices induce chiral automorphisms of the flat 3torus which preserve the linear structure. Accordingly (and echoing the terminology for embedded nets) it is natural to define two graph knots on the same flat torus to be orientedly (or chirally) affinely equivalent if they have translates which correspond to each other under such an N one could associate its primitive graph knot, on the understanding that it is only determined up to oriented affine equivalence.
Thus, to each linear 3periodic netRemark 5.7
We remark that triply periodic surfaces may be viewed as periodic extensions of compact surfaces on the flat 3torus. It follows that the tilings and triangulations of these compact surfaces generate special classes of linear 3periodic nets. Such nets have been considered, for example, in the context of periodic hyperbolic surfaces and minimal surfaces, where the methods of hyperbolic geometry play a role in the definition of isotopy classes (Evans et al., 2013; Hyde et al., 2003; see also Hyde & DelgadoFriedrichs, 2011).
6. Isotopy equivalence
Consider the following informal question: when can N_{0} be deformed into N_{1} by a continuous path with no edge crossings?
This question is not straightforward to approach for two reasons. Firstly, a linear periodic net may contain, as a finite subnet, a linear realization of an arbitrary knot or link. For example, the components of N could be translates of a linear realization of an arbitrary finite knot where all vertices have degree 2. (Here N would have dimension type {3; 0}.) Thus, resolving the question by means of discriminating invariants is in general as hard a task as the corresponding one for knots and links. Secondly, the rules for such deformation equivalence need to be decided upon, and, a priori, the deformation equivalence classes are dependent on these rules.
The following definition may be regarded as the natural form of isotopy equivalence appropriate for the category of embedded periodic nets in 3D which have line segment bonds, no crossing edges and no coincidences of node locations (node collisions).
Definition 6.1
Let N_{0} and N_{1} be proper linear 3periodic nets in . Then N_{0} and N_{1} are periodically isotopic, or have the same periodic isotopy type, if there is a family of such (noncrossing) nets, N_{t}, for 0 < t < 1, for which
(a) there is a continuous path of bases of , , 0 ≤ t ≤ 1, where is a righthanded periodicity basis for N_{t},
(b) there are bijective functions f_{t}:N_{0}→N_{t}, for 0 ≤ t ≤ 1, which map nodes to nodes, such that
(i) f_{0} is the identity map on N_{0},
(ii) for each node point p in N_{0} the map t → f_{t}(p) is continuous,
(iii) the restriction of f_{t} to each edge [a, b] is the unique affine map onto the image edge, [f_{t}(a), f_{t}(b)] in N_{t} determined by linear interpolation.
We make a number of immediate observations:
(1) The condition (iii) could be omitted but is a conceptual convenience in that it implies that each map f_{t} from the body of N_{0} to the body of N_{t} is determined by its restriction to the nodes.
(2) The definition applies to entangled nets with several connected components and in this case the isotopy can be viewed as a set of n independent periodic isotopies, for the n components, with the same time parameter t and periodicity bases , and subject only to the noncollision of components for each value of t.
(3) Every such net N_{0} is periodically isotopic to a model net N_{1} = M with periodicity basis . Indeed, for any righthanded periodicity basis for N there is an elementary isotopy equivalence from to a unique pair which is determined by a path of transformations from which in turn is determined by any continuous path of bases from to .
(4) If N_{0} and N_{1} are orientedly affine equivalent then they are periodically isotopic since the topological group is pathconnected.
We also define the pair to be strictly periodically isotopic to the pair if there is an isotopy equivalence , as in parts (a), (b) of the definition with and . In view of the previous observations we have the following:
Equivalent definition. The embedded periodic nets N and N^{′} in are periodically isotopic if there is a rescaling and rotation of N^{′} to a net N^{′′} so that (i) N and N^{′′} have a common embedded translation group with basis , and (ii) and are strictly periodic isotopic.
Strict periodic isotopy is evidently an equivalence relation on the set of pairs . Periodic isotopy is also an equivalence relation but this is not so immediate. However, as the next proof shows, one can replace a pair of given periodic isotopies, between N_{0} and N_{1} and between N_{1} and N_{2}, by a new pair such that the paths of periodicity bases can be concatenated, and so provide an isotopy between N_{0} and N_{2}.
Theorem 6.2
Periodic isotopy equivalence is an equivalence relation on the set of proper linear 3periodic nets.
Proof
Let be an isotopy equivalence for N_{0},N_{1} as above and let be an isotopy equivalence between N_{1} and N_{2}. Suppressing the implementing maps f_{t} and g_{t} we may denote this information as
We now have two periodic structures and on N_{1}. If they were the same then a periodic isotopy between N_{0} and N_{2} could be completed by the simple concatenation of these paths. However, in general we must choose new periodic structures to achieve this.
For a periodic structure basis and let us write for {k_{1}e_{1}, k_{2}e_{2}, k_{3}e_{3}}. We have for some primitive periodic structure basis of N_{1}. Similarly for some primitive periodic structure basis . Since primitive righthanded periodicity bases on the same linear periodic net are equivalent by a linear map , it follows that the vectors of are integral linear combinations of the vectors of . Thus the vectors of are integral linear combinations of the vectors of . It follows that we can now find elements so that the vectors of are integral linear combinations of the vectors of .
Consider now the induced isotopy equivalences
These isotopies are identical to the previous isotopy equivalences at the level of the paths of individual nodes, but the framing periodic structure bases have been replaced. These periodic isotopies do not yet match, so to speak, but we note that the second isotopy equivalence implies an isotopy equivalence from to some whenever the periodic structure basis has vectors which are integer combinations of the vectors of . Thus we can do this in the case to obtain matching isotopy paths, in the sense that the terminal and initial periodic structure bases on N_{1} agree. Composing these paths we obtain the desired isotopy equivalence between N_{0} and N_{2}.
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6.1. Isotopy equivalence for linear graph knots
In the next definition we formally define two linear graph knots on the flat torus to be linearly isotopic if there is a continuous path of linear graph knots between them. It follows that if the linear graph knots and are linearly isotopic then, by simple periodic extension, the nets N_{1} and N_{2} are periodically isotopic. Also we see in Proposition 6.5 a form of converse, namely that if N_{1} and N_{2} are periodically isotopic then they have graph knots, associated with some choice of periodic structures, which are linearly isotopic.
On the other hand, note that a linear 3periodic net N in with the standard periodicity basis {b} is periodically isotopic to its image N^{′} under an isometric map which cyclically permutes the coordinate axes. This is because there is a continuous path of rotation maps of from the identity map to the cyclic rotation and restricting these maps to N provides maps (f_{t}) for a periodic isotopy. While the associated graph knots and , considered as knots in the same 3torus, are homeomorphic (under a cyclic of the 3torus which maps one graph knot to the other) they need not be linearly isotopic. This follows since linear isotopy within a fixed 3torus must preserve the homology classes of cycles and yet K may contain a directed cycle of edges with a homology class in which do not appear as a homology class of any cycle of edges in K′.
In view of this, in the next formal definition we also give weaker forms of linear isotopy equivalence which can be considered as linear isotopy up to rotations and linear isotopy up to affine automorphisms.
Let . Then there is an induced homeomorphism of the flat 3torus which we denote as X_{π}. This is affine in the sense that flat torus line segments map to flat torus line segments.
Definition 6.3
Let K_{0} = (N_{0}, S_{0}) and K_{1} = (N_{1}, S_{1}) be linear graph knots on the flat torus .
(i) K_{0} and K_{1} are linearly isotopic if there are linear graph knots K_{t} = (N_{t}, S_{t}), for 0 < t < 1, and bijective continuous functions f_{t} : K_{0} → K_{t} such that f_{0} is the identity map on K_{0}, f_{t}(N_{0}) = N_{t}, f_{t}(S_{0}) = S_{t}, and the paths t → f_{t}(p), for p ∈ K_{0} and 0 ≤ t ≤ 1, are continuous.
(ii) K_{0} and K_{1} are rotationally linearly isotopic if for some rotation X_{π}, with X a rotation in , the graph knots K_{1} and X_{π}K_{2} are linearly isotopic.
(ii) K_{0} and K_{1} are globally linearly isotopic if for some affine X_{π}, with , the graph knots K_{1} and X_{π}K_{2} are linearly isotopic.
6.2. Enumerating linear graph knots and embedded nets
We can indicate a linear graph knot K on the flat 3torus by the triple (Q, h, p), where (Q, h) is a labelled directed quotient graph and p = (x_{1}, …, x_{n}) denotes the positions of its n vertices in the flat 3torus . We may also define general placements of K, or of (Q, h), as triples (Q, h, p′) associated with points p′ in the nfold . Such placements either correspond to proper linear graph knots with the same LQG, or are what we shall call singular placements, for which the nodes x_{i}′ of p′ may coincide, or where some pairs of line segment bonds determined by (Q, h) and p′ are not essentially disjoint.
The general placements of K are thus parametrized by the points x′ of the flat 3ntorus , and this manifold is the disjoint union of the set K(Q,h) of proper placements and the set S(Q,h) of singular placements.
Theorem 6.4
There are finitely many linear isotopy classes of linear graph knots in the flat torus with a given LQG.
The following short but deep proof echoes a proof used by Randell (1998) in connection with invariants for finite piecewise linear knots in . However, we remark that an alternative more intuitive proof of this general finiteness theorem could be based on the fact that the isotopy classes of the linear graph knots can be labelled by finitely many crossing diagrams (appropriate to the 3torus). Also direct arguments are available to show such finiteness for LQGs with one or two vertices.
Proof
It suffices to show that there are finitely many representative linear graph knots (with the same given LQG) so that any linear graph knot (with the given LQG) is linearly isotopic to one of them. The set S(Q,h) is a closed semialgebraic set since it is defined by a set of polynomials and inequalities. The open set K(Q,h) is equal to . Since this set is the difference of two algebraic sets it follows from the structure of real algebraic varieties (Whitney, 1957) that the number of connected components of K(Q,h) is finite. Taking a representative linear graph knot from each of these components completes the proof.
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The theorem implies that the isotopy classes of linear graph knots are countable, since LQGs are countable, and so in principle these classes may be listed by various schemes. For example, for each n there are finitely many LQGs of depth 1 with n vertices and so there are finitely many linear isotopy classes of linear graphs knots with n vertices and depth 1.
The corollary of the next elementary proposition gives a similar finiteness for the periodic isotopy classes of embedded periodic nets.
Proposition 6.5
Let N and N^{′} be linear 3periodic nets in . Then the following are equivalent. (i) N and N^{′} are periodically isotopy equivalent. (ii) There are righthanded periodicity bases and for N and N^{′} such that the linear graph knots and are linearly isotopic.
Proof
Suppose that (i) holds. Let N_{0} = N and N_{1} = N^{′} and assume the equivalence is implemented, as in the definition of periodic isotopy, by a path of intermediate nets N_{t} together with (a) a continuous path of bases , 0 ≤ t ≤ 1, where is a periodicity basis for N_{t}, and (b) bijective functions f_{t} from the set of nodes of N_{0} to the set of nodes of N_{t}. The functions f_{t} necessarily respect the periodic structure. Let = and = . It follows that the resulting path is an isotopy between and .
Suppose that (ii) holds, with and . A linear isotopy equivalence (f_{t}) between K and K′ extends uniquely, by periodic extension, to a periodic isotopy equivalence between N and N^{′}.
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Corollary 6.6
Let (H, λ) be a LQG. Then there are finitely many periodic isotopy classes of linear 3periodic nets N which have the LQG (H, λ) with respect to some periodicity basis.
Proof
Fix a LQG (H, λ). Then a linear 3periodic net N^{′} which has the LQG (H, λ) with respect to some periodicity basis is periodically isotopic to a linear 3periodic net M^{′} which has LQG (H, λ) with respect to the standard basis. It suffices to show that the set of such model nets M^{′} has finitely many periodic isotopy classes. This follows since, by Theorem 6.4, their linear graph knots (for the standard basis) have finitely many linear isotopy types and (as in the proof of the previous proposition) a linear isotopy at the graph knot level determines a periodic isotopy at the level of nets, simply by periodic extension.
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In future work it will be of interest to focus on individual topologies and to determine the finitely many periodic isotopy classes of depth 1. Of particular interest are those with some sense of maximal symmetry over their periodic isotopy class. In fact we formalize this idea in Section 7.2 in connection with homogeneous multicomponent nets.
We now note two basic examples of connected selfentangled nets, which we regard as periodic isotopes of their maximal symmetry embedded nets.
Example 6.7
Selfentangled diamond. The multinode fragment in Fig. 7 shows part of an embedded net, say N, whose topology is dia. That N and the usual maximum symmetry net N_{dia} are not periodically isotopic follows from an examination of the catenation of cycles. Specifically the diagram shows that N has two disjoint 6cycles of edges which are linked. This property does not hold for N_{dia} and so they cannot be periodically isotopic.
Example 6.8
Selfentangled embeddings of cds. The maximal symmetry net N_{cds} (associated with cadmium sulfate) has an underlying periodic net cds with quotient graph H(1, 2, 1). The lefthand diagram of Fig. 8 indicates a linear graph knot for cds and the 3periodic extension of this diagram defines a model embedded net which is periodically isotopic to N_{cds}. To be precise, define this net as the model net M(p_{1},p_{2}) with p_{1} = (1/2, 1/2, 1/2), p_{2} = (1/2, 1/4, 1/2) and with LQG where λ assigns the labels, (0, 0, 1) to the loop edge associated with p_{1}, (1, 0, 0) to the loop for p_{2}, and the labels (0, 0, 0) and (0, 1, 0) to the two remaining edges.
As in Section 3.1, let us now view p_{2} as variable point p′_{2} = (x, y, z) within the semiopen cube [0, 1)^{3}. The positions of p′_{2} together with the LQG define model nets as long as there are no edge crossings. Let O be the set of these positions for p′_{2}. Then, viewed as a subset of [0, 1)^{3} (not as a subset of the flat 3torus) the set O decomposes as the union of five pathconnected components O_{1},…,O_{5}. The set O_{1} is the subset of O with y < 1/2, the set O_{2} is the subset with y > 1/2, x > 1/2, z < 1/2 (the righthand figure of Fig. 8 corresponds to a point in O_{2}), and O_{3} is the subset with y > 1/2, x < 1/2, z < 1/2. The sets O_{4},O_{5} are similarly defined to O_{2},O_{3}, respectively, except that z is greater than 1/2.
Let M_{1},…,M_{5} be representatives for the five pathconnected components. The net M_{1} = M(p_{1},p_{2}) is a model net for N_{cds} while the net M_{2} is a periodic isotope. This can be seen once again by the different catenation properties exhibited. Specifically, M_{cds}^{α} has a 6cycle of edges which is linked to (penetrated by) an infinite linear subnet, while M_{cds} does not have such catenation.
6.3. Entangled nets, knottedness and isotopies
The examples above concern connected selfentangled nets and their connected graph knots on the 3torus and there is a natural intuitive sense in which such nets can be `increasingly knotted' by moving through homotopies to embeddings with an increasing number of edge crossings. However, the linear graph knot association is also a helpful perspective for multicomponent nets whose components are not selfentangled so may be equal to, or perhaps merely isotopic to, their individual maximal symmetry embeddings. In this case there are intriguing possibilities for the nesting of such `unknotted components' and their associated space groups. We address this topic in Sections 7 and 8 as well as the attendant crystallographic issue of formulating a notion of maximal symmetry for such multicomponent nets.
For completeness we note two further forms of isotopy equivalence which will not be of concern to us.
(i) Relaxed periodic isotopy. The notion of periodic isotopy equivalence in Definition 6.1 can be weakened in a number of ways. One less strict form, which one could call relaxed periodic isotopy, omits the condition (a), requiring periodic basis continuity, and so allows a general continuous path of intermediate (noncrossing) periodic nets N_{t}. Since the continuity requirement in (b) of the node path functions (f_{t}) is one of pointwise continuity on the set of nodes N_{0}, it follows that such paths of periodic embedded nets can connect embedded nets that are not periodically isotopic. In particular, one can construct relaxed periodic isotopies which untwist infinitely twisted components (e.g. straightening an entangled double helix to a pair of parallel linear strands).
(ii) Ambient isotopy. The usual definition of ambient isotopy for a pair K_{1}, K_{2} of knots (or links) in requires the existence of a path h_{t} of homeomorphisms of (the ambient space) such that h_{0} is the identity map and h_{1}(K_{1}) = K_{2}. Here, for , we have h_{t}(x) = h(t, x) where is a continuous function. Also, the closed sets K_{t} = h(t, K_{0}), for 0 ≤ t ≤ 1, form a path of knots (or links) between K_{0} and K_{1}.
One may similarly define ambient isotopy for embedded periodic nets (DelgadoFriedrichs & O'Keeffe, 2005). In this case the intermediate closed sets L_{t}, defined by L_{t} = h_{t}(N_{0}), are the bodies of general stringnode nets L_{t}. We recall from Power & Schulze (2018) that a stringnode net N in the Euclidean space is a pair (N, S) of sets (whose respective elements are the nodes and strings of N) with the following two properties. (a) S is a nonempty finite or countable set whose elements are lines, closed line segments or closed semiinfinite line segments in , such that collinear strings are disjoint. (b) N is a nonempty finite or countable set of points in given by the intersection points of strings.
It is natural to impose the further condition that these sets are the bodies of (proper) linear 3periodic nets, and this then gives a definition of what might be termed locally periodic ambient isotopy. In this case the set of restriction maps f_{t} = h_{t}_{N0} define a (stricter form of) relaxed periodic isotopy.
7. Group methods and maximal symmetry isotopes
We now give some useful grouptheoretic perspectives for multicomponent frameworks, starting with the general group–supergroup construction in Baburin (2016) for transitive nets. This method underlies various algorithms for construction and enumeration. In this direction we also define maximal symmetry periodic isotopes in terms of extremal group–supergroup indices of the components. Finally, turning towards generically, or randomly, nested components, we indicate the role of Burnside's lemma in counting all periodic isotopes for classes of shifthomogeneous nets.
7.1. Group–supergroup constructions
Let N = N_{1}∪…∪N_{n} be a linear 3periodic net which is a disjoint union of connected linear 3periodic nets in . Let G be the of N and assume that it acts transitively on the n components of N. Thus N is a transitively homogeneous net, or is of transitive type.
Let g_{1} = id, the identity element of G, and note that for each i = 2, …, n there is an element g_{i} ∈ G with g_{i}N_{1} = N_{i}. Also, let H_{i} ⊂ G be the of elements g with g·N_{i} = N_{i}, for i = 1, …, n.
Proof
The cosets g_{i}H_{1} are distinct, since their elements map N_{1} to the distinct subnets N_{i}. On the other hand, if g ∈ G then gN_{1} = N_{j} for some j and so g_{j}^{−1}gN_{1} = N_{1},g_{j}^{−1}g ∈ H_{1}, g ∈ g_{j}H and gH_{1} = g_{j}H_{1}.
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Write to indicate the G which fixes the node v of N and similarly define the group of an edge e of N.
ofProof
It suffices to show that if g fixes an element (vertex or edge) of N_{i} then gN_{i} = N_{i}. Observe that N_{i} is the maximal connected subnet of N containing the element. Also, for any subnet M the image g·M is connected if and only if M is connected, and so the lemma follows.
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These lemmas feature in the proof of the following theorem (Baburin, 2016).
The significance of this result is that it shows that the construction of a transitivetype entangled net N with a connected component M requires the to be free of mirror symmetries which are not in . In fact this necessary condition is frequently a sufficient condition and this leads to effective constructions of novel entangled nets where these nets have components g_{i}M with multiplicity equal to the index of in .
7.2. Maximal symmetry periodic isotopes
Let N be a multicomponent embedded 3periodic net in with and let N_{1},N_{2},…,N_{i},…,N_{n} be representatives of the equivalence classes of the components of N for the translation of . Also, as in the previous section, let H_{i} be the setwise of N_{i} in . Regarding H_{i} as a of Aut(G(N_{i})) (cf. Section 2.2) we may compute the indices m_{i} = Aut(G(N_{i})):H_{i}. Here we restrict our scope to crystallographic nets N_{i} (Klee, 2004) and therefore the indices are always finite. These indices evidently coincide when is transitive on the components of N and this is our primary focus.
We say that is a maximal symmetry N if the nondecreasing rearrangement of m_{1}, …, m_{n}, which we call the multiindex of , is minimal for the lexicographic order when taken over all groups where N^{′} is periodically isotopic to N. In this case we refer to N as a maximal symmetry periodic isotope and we write for this noting that is only defined for such minimal multiindex embedded nets. We note that a maximal symmetry proper embedding of a multicomponent net need not be unique, as might be already the case for (connected) singlecomponent nets (cf. Section 2.2).
for the periodic isotopy class ofIn the same way one may define maximal symmetry groups for periodic homotopy and one may consider other equivalence relations depending on the matter at hand but these issues shall not concern us here.
We note that a maximal symmetry embedding for periodic isotopy is related to the concept of an ideal geometry of a knot (Evans et al., 2015, and references therein) that is required to minimize some energy function. However, as well as a certain arbitrariness in the choice of energy function and the possibility of overlooking a global minimum, the result of optimization depends on the imposed periodic boundary conditions. Thus the determination of maximal translational symmetry embeddings remains problematic in the search for an ideal geometry of a multicomponent periodic net. In contrast, our definition, being essentially grouptheoretic, aims to capture isotopically intrinsic properties of embedded nets which are independent of such constraints.
Maximizing the symmetry of interpenetrated embedded nets is important for a number of reasons, e.g. to characterize their transitivity properties and to derive possible distortions which might occur in a by examining group–subgroup relations. Furthermore, the knowledge of a maximal symmetry can be used to explicitly construct a deformation path that relates an embedding with maximal symmetry to a distorted embedding N^{′} with higher multiindex. A periodic homotopy path can be constructed relative to a common of and , for example, by interpolating between coordinates, and this path is often crossing free and so a periodic isotopy.
Determination of maximal symmetry is a highly nontrivial task. The only general approach to the problem was proposed by Baburin (2016), based on relations between groups of connected components and a respective HRN. Along these lines maximal symmetry embeddings and their symmetry groups have been determined for ngrids as in Section 8.5.
7.3. Counting periodic isotopy classes by counting orbits
Let us now consider embedded nets N with n components on which the acts transitively. We are interested in calculating the number of periodic isotopy classes for a given topology. In the next section we solve this problem for nfold pcu by reducing the counting to a combinatorial calculation, namely to a calculation of the number of orbits of a finite set of `normalized' npcu nets under the action of a finite group of isometries, where the finite group is generated by cube rotations and shifts.
The method is generally applicable but for a translationally transitive npcu embedding a normalization of N takes a particularly natural form in which the components have integral coordinates. While a normalized net is not uniquely associated with N it turns out that their multiplicity corresponds to the cardinality of an orbit under the finite group action, and so counting the number of orbits gives the count we seek. A standard formula for counting such orbits is given by Burnside's lemma which states the following. Let G be a finite group acting on a finite set X with group action x → g · x, so that the orbit of an element z ∈ X is the set {g · z : g ∈ G}. Then the number of distinct orbits is given by
where X_{g} denotes the set of points x with g · x = x. In this way the problem is reduced to counting, for each g, the number of normalized nets which have this symmetry.
8. Classifying multicomponent entangled nets
We next determine the number of periodic isotopy types of various families of embedded nets (linear 3periodic nets) in whose components are embeddings of the net pcu. The simplest family here consists of those nets N with n parallel components, each being a shifted copy of the model net M_{pcu}. In this case we refer to N as a multigrid or ngrid. Such nets have dimension type {3; 3} and are shifthomogeneous.
For practical purposes, both in this section and in Section 9, we focus on the following hierarchy of four equivalence relations for embedded nets:
(1) Nets N_{0},N_{1} are affinely equivalent (respectively, orientedly affine equivalent) if they have translations N_{0}^{′},N_{1}^{′} with N_{0}^{′} = XN_{1}^{′} for some invertible 3 × 3 matrix X (respectively, with ).
(2) The pairs , with given periodicity bases, are strictly periodically isotopic if there is a continuous path of embedded nets N_{t} with an associated continuous path of periodicity bases from to .
(3) N_{0},N_{1} are periodically isotopic if they have strictly periodically isotopic pairs for some choice of periodicity bases .
(4) N_{0},N_{1} are topologically isomorphic, or have the same topology, if their structure graphs (underlying nets) are isomorphic as countable graphs.
8.1. Translationtransitive ngrids
We first consider embeddings of ngrids with a strong form of Specifically we give group–supergroup methods which determine the periodic isotopy types of translationtransitive ngrids.
Considering the translationtransitivity assumption, it follows that the shift vectors relating parallel copies of a singlecomponent grid are in fact ngrids, by enumerating superlattices of index n for the lattice of a connected component while discarding the associated ngrids which fail to be noncrossing.
representatives of some lattice with respect to the generated by the standard periodicity basis of a connected component. The number of cosets is equal to the index of a This observation gives a recipe for generating translationtransitiveA determination of indexn superlattices can be made with the following lemma [see also Cassels (1997), Davies et al. (1997)].
Lemma 8.1
Let n have a factorization n = p_{1}p_{2}p_{3}, with 1 ≤ p_{i} ≤ n, and let
be a matrix with integral entries satisfying 0 ≤ q_{1} < p_{2}, 0 ≤ q_{2} < p_{3} and 0 ≤ r_{1} < p_{3}. The rows of the inverse matrix L^{−1} generate a of of index n. Moreover, every of of index n has such a representation.
A computational determination of the number, β_{tt}(n), of periodic isotopy types can now be implemented with the following threestep algorithm. Some of the values are recorded in the summary Table 1. (i) Using the lemma, generate all superlattices with index n. (ii) Discard such a if its corresponding ngrid has edge crossings. (iii) Reduce the resulting list to a (maximal) set of superlattices which are pairwise inequivalent under the of a primitive cubic lattice.
We have indicated that this (practical) threestep generationandreduction algorithm gives the number of congruence classes of translationally transitive ngrids. That this number also agrees with the (a priori smaller) number of periodic isotopy classes (up to chirality) is essentially a technical issue. This follows from Theorem 8.2 (iii) and Appendix A. Moreover, for the same reason the algorithm determines exactly the translationally transitive ngrids which are maximal symmetry periodic isotopes.
We remark that a similar threestep algorithm can be applied in the case of translationally transitive embeddings of e.g. nfold dia, nfold srs and other nets. We conjecture that if connected components are crystallographic nets in their maximal symmetry configurations, then step (iii) leads directly to the classification into periodic isotopy classes.
8.2. A combinatorial enumeration of ngrids
We now consider the wide class of general multigrids, with no further symmetry assumptions. The combinatorial objects relevant to periodic isotopy type counting are given in terms of various finite groups acting on finite sets of patterns which we now define.
Let T = {1, …, n}^{3}, viewed as a discrete 3torus, and let C_{n} be the cyclic group of order n. In particular C_{n} can act on T by cyclically permuting one of the three coordinates. Also, let R be the rotation symmetry group of the unit cube [0, 1]^{3}. Then R acts on the discrete torus T in the natural way.
Let X(n) be the finite set of unordered ntuples, or patterns, {p_{1}, …, p_{n}} where the points lie in T and have distinct coordinates, so that for all pairs p_{i}, p_{j} the difference p_{i} − p_{j} has nonzero coordinates. In particular X(n) = (n!)^{2}. These ntuples in fact correspond to the coordinates of the nodes appearing in a of the n components of a normalized ngrid.
Finally, for , let ρ(n), α(n), β(n), respectively, be the number of orbits in X under the natural action of the groups
Recall that a linear graph knot for an embedded net N is determined by a choice of periodicity basis and is denoted . In the case of an ngrid N with its standard periodicity basis we refer to as the standard linear graph knot for N. Evidently appears as the union of n disjoint translates in the flat 3torus of K_{pcu}.
Theorem 8.2
(i) The number of linear isotopy types of standard linear graph knots of ngrids is α(n). (ii) The number of rotational isotopy types of standard linear graph knots of ngrids is β(n). (iii) The number of periodic isotopy classes of ngrids is β(n).
The proof of this theorem is given in Appendix A. The essential argument involves a discretization in which, in (ii) for example, the components are separately shifted by a (joint) isotopy to an evenly spaced position. Then n nodes in a correspond to a pattern of n coordinate distinct points in the discrete torus {1, 2, …, n}^{3}. Additionally, for (iii) one must resolve the technical problem in Remark 11.1 in the case of ngrids and show that the triple cyclic order of coordinates (modulo the rotation group R) is indeed a periodic isotopy invariant. We do this in Lemma 11.2, and the equivalence given in Proposition 6.5 is a helpful step in the proof. We also note that the periodic isotopy that one needs to construct in the proof, when the cyclic orders coincide modulo R, is simply a concatenation of a periodic isotopy of local component translations (to achieve equal spacing), followed by an elementary periodic isotopy induced by a path of affine motions corresponding to a (bulk) rotation and final translation.
8.3. Translational isotopy and framed ngrids
The general formulation of periodic isotopy of necessity entails some technical complexity in the proofs. We now note two restricted but natural ngrid contexts where the determination of the number of equivalence classes simplifies. We omit the formal proofs. In the first of these we define a more restricted form of isotopy while in the second context we distinguish, or colour, one of the component grids.
Let us say that a multigrid is aligned if its components M_{i} are translates of the model net M_{pcu} with node set .
Definition 8.3
Two aligned ngrids M,M^{′} are translationally isotopic if for some labelling of the components there are continuous functions , for 1 ≤ i ≤ n, with g_{i}(0) = 0 for all i, such that
(i) for each t the embedded net
is a (noncrossing) linear 3periodic net,
(ii) M = M(0) and M^{′} = M(1).
This simple form of the periodic isotopy t→M(t) in fact corresponds to strict periodic isotopy for these nets with respect to the standard periodicity basis. It is a form of `local' periodic isotopy in the sense that the deformation paths of the nodes are localized in space. In particular deformation paths incorporating bulk rotations are excluded.
For the second variation, let us define a framed ngrid to be an (n + 1)grid with a distinguished component, the framing component. Thus a framed ngrid is a coloured n + 1 grid where all but one of the components are of the same colour. Periodic isotopy for coloured ngrids may be defined exactly as before but with the additional requirement that the maps (f_{t}) respect colour.
It is evident that the cube rotation group R acts naturally on such framed ngrids. Also, as indicated in our remarks following Theorem 8.2, counting periodic isotopy types reduces to counting orbits of patterns p of n + 1 coordinatedisjoint points, p = (p_{1}, …, p_{n+1}), in the discrete torus {1, 2, …, n + 1}^{3}. However, in view of the colour preservation we may assume, by shifting, that p_{n+1} lies in the Rorbit of (1, …, 1), and from this it follows (varying the proof of Theorem 8.2) that the periodic isotopy classes correspond to the Rorbits of the ntuples (p_{1}, …, p_{n}).
8.4. Employing Burnside's lemma
We can now make use of Burnside's lemma to compute values of α(n), β(n) and ρ(n). The following formula readily shows that α(5) = 128, α(7) = 74088 for example.
Proof
Note that a group element g = abc ≠ 000 in C_{p} × C_{p} × C_{p} with a or b or c equal to 0 does not fix any pattern under the cyclic action on T = {1, 2, …, p}^{3} and so X_{abc} = 0 in this case. Also, every pattern is fixed by the identity element and so X_{000} = (p!)^{2}. It remains to consider the (p − 1)^{3} group elements abc with none of a, b, c equal to 0.
The group element 111 acts as a diagonal shift and so any fixed pattern of nodes in T is determined by the unique node occupying a particular face of T. Conversely any of the p^{2} node locations on this face determines a unique fixed pattern for the action of 111. Thus X_{111} = p^{2}.
Since p is prime the same argument applies to any group element abc with none of a, b, c equal to the identity element 0, since a, b, c each have order p. There are (p − 1)^{3} such elements abc and so the formula now follows from Burnside's lemma.
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The case of composite n is similar. In the case that each of a, b, c have order r where r divides n the size of the fixed set X_{g} for g = abc is the product n^{2}(n−r)^{2}(n−2r)^{2}…r^{2}. All other elements except the identity have no fixed patterns. In this way we obtain α(4) = 12 and α(6) = 2424.
Similarly, for the framed ngrids one may compute ρ(2) = 1, ρ(3) = 4, ρ(4) = 30. Evidently there is a rapid subsequent growth rate since the Burnside lemma formula quickly leads to the lower bound ρ(n) ≥ (n!)^{2}/(24 × n^{3}).
8.5. Classes of embedded npcu
Fig. 9 gives examples of small ngrids with contrasting transitivity properties. For more details, see the supporting information.
In Table 1 we summarize the number of classes of ngrids for various types of ngrid and forms of isotopy for some small values of n with the values of α(n) and β(n) obtained via Burnside's lemma as before. The count β_{t}(n) is for transitive ngrids in the sense given in Section 2.4, and for ngrids this coincides with vertex transitivity. The count β_{tt}(n) is for translationtransitive ngrids which have components that are equivalent by translations in the These counts, which coincide if n is prime, are obtained using the group–supergroup algorithm of Section 7.2.
Fig. 10 summarizes and transitivity types of multicomponent embedded nets.
9. Classifying lattice nets
9.1. Depth1 disconnected nets with a singlevertex QG
A model net M which has adjacency depth 1 with respect to the standard basis is determined by a set F_{e} of edge representatives [a, b] for the translational orbits of edges. In the case that there is a single orbit for the nodes we may assume that there is a node at the origin and choose the unique edgeorbit representative [a, b] such that (a, b) is a subset of the semiopen cube [0, 1)^{3}. Such representative edges are determined up to sign by the vectors a − b, or equivalently in this case, by the labels of the depth1 . We use the following terminology for edges in F_{e}. This will also be useful in subsequent sections.
The three axial edges are denoted a_{x}, a_{y}, a_{z} and d_{1}, …, d_{4} denote the four diagonal edges which are incident to (0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0), respectively. The three face diagonal edges which are incident to the origin are denoted f_{x}, f_{y}, f_{z}, corresponding to the directions (0, 1, 1), (1, 0, 1), (1, 1, 0), while the edges g_{x}, g_{y}, g_{z} are the other face diagonals, parallel to the vectors (0, 1, −1), (1, 0, −1), (1, −1, 0), respectively. Thus we may define any set F_{e} by means of an ordered subword w of the ordered word
In view of the noncrossing condition it is elementary to see that every model net M is affinely equivalent, simply by rotations, to a standardized model net defined by the standard ordered word of the form w = w_{1}w_{2}d_{1} or w_{1}w_{2}, where w_{1} is either a_{x}, a_{x}a_{y}, a_{x}a_{y}a_{z} or the null word, and w_{2} is a face subword with zero, one, two or three letters, of which there are 27 possibilities.
We now determine the depth1 embedded bouquet nets that are disconnected, that is, which have more than one and possibly infinitely many connected components. It turns out that there are six embedded nets up to affine equivalence and we now give six model nets for these types.
(i) M_{a} is the model net determined by F_{e} = {a_{x}} and consists of parallel copies of a 1periodic linear subnet.
(ii) M_{aa} is determined by the word a_{x}a_{y} and is the union of parallel planar embeddings of sql.
(iii) M_{aafz} is the net for a_{x}a_{y}f_{z} and is the union of parallel planar embeddings of hxl.
(iv) M_{fff} is the net for f_{x}f_{y}f_{z} and is the translationtransitive union of two disjoint copies of an embedding of pcu.
(v) M_{ggd} is the net for g_{x}g_{y}d_{1} and is the translationtransitive union of three disjoint copies of an embedding of pcu.
(vi) M_{ggg}^{d} is the net for g_{x}g_{y}g_{z}d_{1} and is the translationtransitive union of three disjoint copies of an embedding of hex.
In the above list, and in Tables 1, 3, we use a compact notation where the letter subscripts for the nondiagonal edges are suppressed if they appear in alphabetical order, and where d indicates the diagonal edge d_{1}. Thus, the model net for w = g_{x}g_{y}g_{z}d_{1}, which could be written as M(g_{x}g_{y}g_{z}d_{1}), is written in the compact form M_{ ggg}^{d}. Its repeating unit, or motif, is indicated in Fig. 11 along with a fragment of the embedding rotated so that the penetrating edges are vertical.
Theorem 9.1
There are six affine equivalence classes of disconnected embedded nets with adjacency depth 1 and a singlevertex quotient graph.
Proof
Let M be a model net of the type stated, with generating edge set F_{e} with F_{e} = m. If m = 1 (respectively, m = 2) then M is affinely equivalent to M_{ a} (respectively, M_{ aa}).
Let m = 3. Then the three edges of F_{e} have separate translates, under the periodic structure, to three edges in M which are incident to a common node. Suppose first that this triple is coplanar. Then it determines a planar subnet, M_{1} say, which is an embedding of hxl. Also M is equal to the union of the translates of M_{1} of the form M_{1}+nb where b is a vector of integers and . Thus M is affinely equivalent to M_{ aaf}.
On the other hand, if the edges of F_{e} are not coplanar then M_{1} is an embedding of pcu. Examination shows that this occurs with M disconnected, only for words w of the forms (i) fff, giving two components, (ii) fgg, gfg or ggf, each of which is of type fff after a translation and rotation, and (iii) ggd, which gives three components.
For m ≥ 4 the model net M_{ ggg}^{d} is the only net which is not connnected.
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9.2. Connected lattice nets with depth 1
Let denote the family of proper linear 3periodic nets with a periodic structure basis providing a depth1 LQG with a single vertex. We now consider the subfamily of connected nets N in . These nets also give building block nets for embedded nets with a doublevertex quotient graph, and for multicomponent nets. In Theorem 9.5 and Corollary 9.6 we classify the nets up to oriented and up to periodic isotopy, respectively, there being 19 classes for each equivalence relation. As in the previous section it will suffice to consider model nets. Moreover each model net M in is determined by an ordered word for the edges of a repeating unit F_{e} and these edges [a, b] are subsets of the [0, 1)^{3} except for one of their endpoints. In view of connectivity and noncrossing conditions the QG(M) is a bouquet graph with a single vertex and loop edges of multiplicity m = 3, 4, 5, 6 or 7 (as implied by Lemma 4.4).
To distinguish these model nets we make use of some new readily computable local features which can be read off from the repeating unit and which provide some readily computable structural invariants under affine isomorphism.
Definition 9.2
The hxlmultiplicity hxl(N) of an embedded net N is the number of translation classes of planar 2periodic subnets of N which are completely triangulated.
For the model nets M in this multiplicity is equal to the number of triples of edges [a, b] in F_{e} whose edge vectors, b − a, form a coplanar triple. It may also be computed from the PS as the number of 3cycles divided by 6. Thus the PS of fcu is 3^{24}.4^{36}.5^{6} and so hxl(N_{fcu}) = 4.
The next definition might be viewed as a strong form of local catenation.
Definition 9.3
(i) An edge of an embedded net is 3^{2}penetrating if there exist two disjoint parallel edgecycles of length 3 and an edge [a, b] which passes through them in the sense that the open line segment (a, b) intersects the convex hull of each cycle. (ii) An edge of an embedded net is 4^{2}penetrating if it passes through two disjoint parallel untriangulated parallelograms.
One can check for example that for the model net M in with a defining ordered word w there exists a 3^{2}penetrating edge if and only if w contains the subword g_{x}g_{y}g_{z}d_{1}. Also there exists a 4^{2}penetrating edge if and only if w contains d_{1} and precisely two of the three letters g_{x}, g_{y}, g_{z}. See Fig. 12.
We similarly define when an edge is 3^{1}penetrating or 4^{1}penetrating. In fact there are no depth1 lattice nets with a 3^{1}penetrating edge. In general let us say that N has property 3^{k} if there are 3^{k}penetrating edges but no 3^{k+1}penetrating edges. We also define property 4^{k} similarly. We indicate these properties in column 5 of Table 3.
9.3. Classification of depth1 lattice nets
We now define 19 model nets M in with standard orthonormal basis as a depth1 periodicity basis and where in each case the node set is the subset of . We do this, as in the previous section, by specifying a defining edge word, as listed in column 2 of Table 3. The nine nets without the strong edge penetration property (of type 3^{2} or 4^{2}) appear in the RCSR whereas the other ten nets do not. This reflects the fact that the strongly penetrated nets can be viewed as exotic forms in reticular chemistry. Indeed, there are three new topologies which have not been observed either in the RCSR or the ToposPro net databases. Two of these are provided by the model nets M^{ggg}_{ ad},M^{ggg}_{ aad} given in Fig. 13.
We also record in the final column the cardinality of the π(pcu) etc.
of the maximal symmetry net with the given topology, which we may denote byLet us define an elementary affine transformation of to be a rotation, a translation or a linear map whose representing matrix has entries 1 on the main diagonal and a single nonzero nondiagonal entry equal to 1 or −1. These maps, such as (x, y, z) → (x − z, y, z), map model nets to model nets and play a useful role in casebycase analysis.
Remark 9.4
We note that the countable graph ilc, represented by the model net M_{ aad}^{g}, can be represented in other ways. The model net M_{ fff}^{d} gives one such alternative. The topology is also made apparent by its equivalence, by elementary transformations, with the net obtained from the pcu model net by the addition of integer translates of the long diagonal edges with edge vector (1, 2, 1). However, in this case the standard basis is a periodic structure basis of depth 2.
Theorem 9.5
There are 19 oriented affine equivalence classes of connected lattice nets with depth 1.
We have obtained this classification by means of a casebycase proof as well as a verification by an enumeration of lattice nets using GAP. The following interesting special case, with two new nets, illustrates the general proof method. (See the supporting information for the complete proof.)
Determination of the 10coordinated connected lattice nets of depth 1
Suppose first that a model net M in this case has three axial edges and two face edges. Then it is straightforward to see that it is equivalent by elementary affine transformations to the model net M_{pcu}^{ff}, for bct. Also, any model net of type aaafd is similarly equivalent to this type. On the other hand, a type aaagd model net has hxlmultiplicity equal to 1, rather than 2, and so represents a new equivalence class. Its topology is ile.
Consider next the model nets with two axial edges and no diagonal edges. These are equivalent by elementary affine transformations to a model net with three axial edges and so they are equivalent to the model nets in Table 3 for bct and ile. The same is true for the nine nets of type aawd where w is a word in two facial edges which is not of type gg.
Thus, in the case of two axial edges it remains to consider the types a_{x}a_{y}wd_{1} with w = g_{x}g_{y}, g_{x}g_{z} and g_{y}g_{z}. Each of these has a penetrating edge of type 4^{2}. The first two are model nets in the list and they give new and distinct affine equivalence classes in view of their penetration type and differing hxl(N) count. The third net, for the word a_{x}a_{y}g_{y}g_{z}d_{1}, is a mirror image of the first net and is orientedly affinely equivalent to it, by Remark 5.5 for example.
It remains to consider the case of one axial edge, a_{x}, together with d_{1} and three facial edges. If there are two edges of type f_{x}, f_{y} or f_{z} then there is an elementary equivalence with a model net with two axial edges. The same applies if there is a single such edge. [For an explicit example consider a_{x}f_{x}g_{y}g_{z}d_{1} and check that the image of this net under the transformation (x, y, z) → (x, y − z, z) gives a depth1 net with two axial edges.]
Finally the model net for a_{x}g_{x}g_{y}g_{z}d_{1} appears in the listing and gives a new class with penetration type 3^{2}.
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Corollary 9.6
There are 19 periodic isotopy classes of connected linear 3periodic nets in with adjacency depth 1 and a singlevertex quotient graph.
Proof
If the connected linear 3periodic nets N_{1},N_{2} are orientedly affinely equivalent then, as previously observed, they are periodically isotopic. Thus there are at most 19 periodic isotopy classes. On the other hand, periodically isotopic embedded nets have structure graphs which are isomorphic as countable graphs. Since the 19 model nets have nonisomorphic structure graphs the proof is complete.
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Theorem 9.5, together with the linear implementation of graph isomorphisms indicated in Theorem 4.8, implies that the structure graphs of the 19 model nets must be nonisomorphic as graphs. This also follows on examining the td10 topological density count. We remark that Table 3, without the final td10 column, almost distinguishes 19 affine equivalence classes (and hence, by Theorem 4.8, the structure graphs) since we have only appealed to topology density to distinguish the curious pair M_{ ad}^{gg} (8T17), M_{ ad}^{gygz} (8T21). Fig. 14 shows the maximal symmetry embeddings of the nine model nets of which do not have the 3^{2} or 4^{2}penetration property.
10. Double lattice nets and further directions
We give a brief indication of research directions in the determination of periodic isotopy classes and periodic isotopes for embedded nets with a doublevertex quotient graph as well as research directions in rigidity and flexibility.
10.1. Double lattice nets
For convenience we define a double lattice net to be an embedded periodic net N in whose set of nodes is the union of two translationally equivalent rank3 lattices and we let be the family of proper double lattice nets with adjacency depth 1.
The doublevertex quotient graph N in consists of two bouquet graphs and a number of nonloop edges. We denote these graphs as H(m_{1}, m_{2}, m_{3}) where m_{1} and m_{3} are the loop multiplicities, with m_{1} ≥ m_{3} ≥ 0, and m_{2} is the multiplicity of the connecting edges. From Lemma 4.4 we have the necessary conditions 0 ≤ m_{1} ≤ 7 and 0 ≤ m_{2} ≤ 8 as well as m_{3} ≥ 1 if m_{2} = 1, since, from the definition of a linear 3periodic net, there can be no nodes of degree 1. If N is a net in , the subfamily of connected nets, then we also have the additional condition m_{2} ≥ 1.
Each net admits a unique threefold decomposition N = N_{1}∪N_{2}∪N_{3} where N_{1} and N_{3} are the disjoint 3periodic subnets associated with the two bouquet subgraphs and where N_{2} is the net, with the same node set as N, associated with the subgraph with nonloop edges. The subnets N_{1},N_{3} may have no edges if one or both vertices has no loop edges. When loops are present on both vertices then the nets N_{1},N_{3} are bouquet nets, and are of three possible dimension types, namely {3; 1}, {3; 2} or {3; 3}. As we have seen earlier, for type {3; 1} there is one class of embedded nets, for type {3; 2} there are two such classes and for type {3; 3} there are three classes for disconnected nets and 19 classes for connected nets.
Thus in the threefold decomposition of a net N in , each of the subnets N_{1}, N_{3} is either devoid of edges or is separately orientedly affinely equivalent to one of the 25 model nets for . The relative position (parallel or inclined, for example) of these component nets allows for considerable diversity for the entangled net N_{1}∪N_{3}. In particular, while N is affinely equivalent to a general model net M_{1}∪M_{2}∪M_{3}, with standard periodic structure basis , in general we can only additionally arrange that one of the subnets M_{1},M_{3} is equal to a translate of one of the specific 25 model nets in Tables 2 and 3.
Evidently there is a considerably diversity for the periodic isotopy classes of embedded nets with depth 1 and a doublevertex quotient graph. We now show that there is even a marked increase in the number of topologies for such nets.
For 1 ≤ m ≤ 8 define to be the family of nets N in which have a periodicity basis with a depth1 bipartite quotient graph H(0, m, 0) with an edge carrying the label (0, 0, 0). The label condition here ensures the natural condition that N has an edge between the pair of representative joints in the semiopen for the periodicity basis. In fact this convention, which we call the unitcell property, is the natural convention used by Chung et al. (1984) in their schemes for the enumeration of periodic nets.
For m = 1, 2, 3 the nets of this type are not connected. For m = 4 it is well known that there is a unique connected topology G(N) for the nets in , namely the diamond net dia (Beukemann & Klee, 1992). For higher values of m we are able to determine the topologies through a computational analysis based in part on the indivisibility criterion Proposition 4.5. See also the supporting information.
10.2. Rigidity and flexibility
The analysis of infinitesimal rigidity and flexibility for connected crystal frameworks C is a welldeveloped mathematical topic. In its simplest form a velocity field on the node set is assumed to be periodic with respect to a given periodicity basis . This is the socalled fixed lattice theory and in fact it corresponds exactly to the rigidity theory of fixed edgelength graph knots on a fixed flat torus for the parallelepiped defined by the periodicity basis. In this case a finite matrix, the periodic rigidity matrix for the pair , determines the space of periodic infinitesimal flexes and so this matrix is a discriminator for the (strict) periodic rigidity of C with respect to . On the other hand, the flexible lattice theory allows for infinitesimal motions of the periodicity basis and so embraces a larger finite dimensional of velocity fields with a correspondingly larger rigidity matrix (see Borcea & Streinu, 2010; Power, 2014b). Recently, necessary and sufficient conditions have been given for infinitesimal rigidity with respect to the infinite dimensional space of all velocity fields (see Kastis & Power, 2019).
The fixed lattice theory also has close connections with the analysis of rigid unit modes (RUMs) in material crystals with a connected bondnode net. See for example the RUM mode analysis in the work of Badri et al. (2014), Power (2014a). In fact this analysis also applies to disconnected crystal frameworks with several components if there are no interaction constraints between the components. Indeed, suppose that C belongs to the interpenetration class and let be a periodicity basis for both C and each of its finitely many components C_{i}. Then the RUM spectrum Ω(C) of C, with respect to , is the union of the RUM spectra of its components.
A crystal framework is said to be critically coordinated, or to be a Maxwell framework, if the quotient graph satisfies E = 3V. This is often interpreted as an equality between the total number of constraints (provided by E equations) that restrict the total number of of a repeating unit of nodes, which is 3V. It also implies an equality of limits of averages over increasing volumes for these constraint/freedom quantities. It is for such frameworks, which includes all zeolite frameworks for example, that the RUM spectrum is typically a nontrivial algebraic variety exhibiting detailed structure (Dove et al., 2007; Power, 2014a; Wegner, 2007).
In the light of this it is of interest to determine the basic Maxwell frameworks C which have a depth1 LQG with either one or two vertices. From Proposition 10.1 it follows that there are 31 topologies for crystal frameworks of this type with the unitcell property and quotient graph H(0, 6, 0). These remarks suggest that it would be worthwhile to augment periodic net database resources with tools for the identification of Maxwell lattices and the calculation of flexibility information related to RUM spectra.
APPENDIX A
Proof of Theorem 8.2
Note first that any connected component K_{i} of K is determined by the position of its unique node in [0, 1)^{3}. Thus K is determined by the position p_{i} = (x_{i}, y_{i}, z_{i}), 1 ≤ i ≤ n, of its ntuple of nodes. Also, in view of the disjointness of components two such nodes p_{i}, p_{j} have differing corresponding coordinates in [0, 1). Consider a deformation path (f_{t}) from K to K′. Since the graph knots f_{t}(K) are also graph knots of ngrids, and edge collisions cannot occur in the deformation, it follows that the cyclical order of the x, y and z coordinates of the points f_{t}(p_{1}), …, f_{t}(p_{n}), is constant. Thus the ordered triple of cyclic orders for the coordinates is an invariant for linear graph knot isotopy.
Despite the constraint of coordinate distinctness we see that K can be linearly isotopic to an ngrid graph knot K′ determined by p_{i}′ = (x_{i}′, y_{i}′, z_{i}′), 1 ≤ i ≤ n, where the n coordinates x_{i}′ lie at the midpoints of the distinct subintervals of the form [j/n, (j + 1)/n), 0 ≤ j ≤ n − 1. This spacing is achieved by simultaneously translating the points p_{i} in the x direction at appropriate independent speeds while maintaining xcoordinate distinctness. Additionally, the equal spacing of the y and z coordinates can be achieved by similar isotopies which locally translate in the y and z directions. The resulting position is unique up to the cyclic permutation action of C_{n} × C_{n} × C_{n} on the coordinate axes. It follows now that two graph knots of ngrids are linearly isotopic if the cyclic order of their coordinates coincides. Thus the set of cyclic orders is a complete invariant for linear graph knot isotopy and (i) and (ii) follow.
Assume next that the ngrids N and N^{′} are periodically isotopic. It will suffice to show that their linear graph knots are rotationally linearly isotopic.
Without loss of generality we may assume that the components have node sets that lie on translates of the lattice in . Thus, by the definition of periodic isotopy there are periodicity bases and , with integer entries, such that and are strictly periodically isotopic by means of a deformation path (f_{t}) and an associated path of bases from to . Define k_{1} to be a common multiple of the x coordinates of {a_{1}, a_{2}, a_{3}} and similarly define k_{2}, k_{3} for the y and z coordinates. Then there is an implied periodic isotopy between and , for some periodicity basis with integer entries. This is given by the same periodic isotopy deformation path (f_{t}) but with a new associated path of bases (for lower translational symmetry) which is determined by the initial basis and the path . So, without loss of generality we may assume at the outset that .
We next show that is equal to where k′ is a cyclic permutation of k. Thus we will obtain that the linear graph knots are rotationally linearly isotopic.
To see this consider a single component N_{0}^{1} of the ngrid N_{0}. Note that the linear graph knot has minimal discrete length cycles c_{1}, c_{2}, c_{3} with homology classes δ_{1}, δ_{2}, δ_{3}, respectively, equal to the standard generators of the homology group of the containing flat 3torus. These discrete lengths are k_{1}, k_{2}, k_{3}. Moreover we see, from the rectangular geometry of N_{0}^{1}, the following uniqueness property, that if c_{1} and c′_{1} are two such minimal length cycles for δ_{1} which share a node then c_{1} = c′_{1}. Indeed, the minimality implies that the edges of c_{1} are parallel or, equivalently, that the nodes of c_{1} can only differ in the x coordinate.
Let N_{1}^{1} be the corresponding component of N_{1}. In fact N_{1}^{1} = f_{1}(N_{0}^{1}). The linear graph knot is, by definition, equal to the affine rescaling of the intersection of the body N_{1}^{1} with the semiopen parallelepiped defined by the periodicity vectors a′_{1}, a′_{2}, a′_{3}. We note that if is not of the form then for at least one of the standard generators δ_{i} (associated with a_{i}′) of the flat 3torus homology group , the minimal length cycles do not have the uniqueness property. This follows from elementary geometry since not all edges of the cycle can be parallel when a′_{i} is not parallel to a coordinate axis.
On the other hand, the linear isotopy between and preserves the lengths of cycles of edges and so the claim follows. Since we have shown that and are isotopic linear graph knots up to a rotation, it follows from the technical lemma, Lemma 11.2, that the graph knots and are linearly isotopic up to a rotation, and so (iii) now follows from (ii).
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Remark 11.1
Recall the notation introduced in the proof of Theorem 6.2. Let us say that this denotes the amplification by of the periodic structure basis . We pose the following general problem. If N_{1},N_{2} are linear periodic nets which have a common periodic structure basis and if the pair is periodically isotopic to the pair , then does it follow that N_{1} and N_{2} are periodically isotopic? In view of the proposition above this is equivalent to the corresponding problem for linear graph knots K and their kfold amplifications which we may write as k · K. We expect that this is true and therefore that the amplified knots are isotopic if and only if the unamplified knots are isotopic. In fact one can verify this connection for various specific classes of interest, as we do below in the case of multigrid nets.
The following technical lemma resolves the question of Remark 11.1 in the case of ngrids.
Lemma 11.2
Let N_{0} and N_{1} be shifthomogeneous ngrids with standard linear graph knots K_{0} and K_{1} and suppose that for some the amplified graph knots k · K_{0} and k · K_{1} are isotopic. Then K_{0} and K_{1} are strictly linearly isotopic.
Proof
Fig. 15 indicates a subgraph knot, C_{0} say, of one of the components of k · K_{0} in the case that k = (3, 6, 5). We refer to this as a `chain'. It consists of a small cube of edges attached to three cycles of edges in the axial directions. Let p_{1} denote the vertex which is common to these three cycles. The other n − 1 components of k · K_{0} have similar chains which are shifts of C_{0} and there is a unique such chain where the shift of p_{1} lies in the semiopen small cube p_{1} + [0, 1)^{3} of the flat 3torus. Let p_{2}, …, p_{n} be these axial joints and let J_{0} = J_{0}(p_{1}, …, p_{n}) be the union of these chains (giving a linear subgraph knot of k · K_{0}).
Suppose that k · K_{0} and k · K_{1} are linearly isotopic, by the isotopy (g_{t}), 0 ≤ t ≤ 1. This restricts to a linear isotopy from J_{0} to a subgraph knot g_{1}(J_{0}) of k · K_{1}. In this isotopy the images under g_{t}, for 0 < t < 1, of the n axial cycles of J_{0} in a specific coordinate direction need not be linear. However, since there can be no collisions the cyclical order for t = 0 agrees with the cyclical orders for t = 1. It follows that g_{1}(J_{0}), which has the form J_{0}(q_{1}, …, q_{n}), is a subgraph knot of k · K_{1} of the same cyclical type as the subgraph knot J_{0}. Since K_{0} and K_{1} are also defined by the cyclical order of p_{1}, …p_{n} and q_{1}, …, q_{n} it follows that they are linearly isotopic.
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Supporting information
Proof of Theorem 9.5. DOI: https://doi.org/10.1107/S2053273320000625/ib5087sup1.pdf
Coordinates for nets from Table 3 (in cgd format). DOI: https://doi.org/10.1107/S2053273320000625/ib5087sup2.txt
Coordinates for nets from Section 10 (in cgd format). DOI: https://doi.org/10.1107/S2053273320000625/ib5087sup3.txt
Coordinates for ngrids (corresponding to beta values) from Table 1 (in cgd format). DOI: https://doi.org/10.1107/S2053273320000625/ib5087sup4.txt
Funding information
This work was supported by the Engineering and Physical Sciences Research Council (grant No. EP/P01108X/1). Davide M. Proserpio thanks the Universita degli Studi di Milano for the transition grant PSR20151718 and for the grant FFABR2018.
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