Ideal geometry of periodic entanglements

Author:

Evans Myfanwy E.1,Robins Vanessa2,Hyde Stephen T.2

Affiliation:

1. Department of Mathematics, TU Berlin, Str. des 17. Juni 136, Berlin 10623, Germany

2. Department of Applied Mathematics, Research School of Physics and Engineering, 60 Mills Road, The Australian National University, Acton ACT 2601, Australia

Abstract

Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo . This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ -graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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