### Acta Crystallographica Section A

# Foundations of Crystallography

### Volume 69, Part 3 (May 2013)

## research papers

*Acta Cryst.* (2013). A**69**, 241-261 [ doi:10.1107/S0108767313001670 ]

### Periodic entanglement I: networks from hyperbolic reticulations

**Abstract:** High-symmetry free tilings of the two-dimensional hyperbolic plane () can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane *via* projection onto four genus-3 TPMSs: the *P*, *D*, *G*(yroid) and *H* surfaces. The entanglements of these patterns are explored and partially characterized using tools from *TOPOS*, *GAVROG* and a new tightening algorithm.

**Keywords: periodic entanglement; networks; hyperbolic reticulations; tilings; ideal knots; tight knots; crystal nets; frameworks; surfaces; graphs; catenation; enumeration.**

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