TY - JOUR
T1 - Ideal geometry of periodic entanglements
AU - Evans, Myfanwy E.
AU - Robins, Vanessa
AU - Hyde, Stephen T.
N1 - Publisher Copyright:
© 2015 The Authors.
PY - 2015/9/8
Y1 - 2015/9/8
N2 - 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 è-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.
AB - 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 è-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.
KW - Entanglement
KW - Graphs
KW - Ideal knots
KW - Networks
KW - Weavings
UR - http://www.scopus.com/inward/record.url?scp=84943174337&partnerID=8YFLogxK
U2 - 10.1098/rspa.2015.0254
DO - 10.1098/rspa.2015.0254
M3 - Article
SN - 1364-5021
VL - 471
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2181
M1 - 20150254
ER -