Three-dimensional entanglement: knots, knits and nets

Abstract

Three-dimensional entanglement, including knots, periodic arrays of woven filaments (weavings) and periodic arrays of interpenetrating networks (nets), forms an integral part of the analysis of structure within the natural sciences. This thesis constructs a catalogue of 3-periodic entanglements via a scaffold of Triply-Periodic Minimal Surfaces (TPMS). The two-dimensional Hyperbolic plane can be wrapped over a TPMS in much the same way as the two-dimensional Euclidean plane can be wrapped over a cylinder. Thus vertices and edges of free tilings of the Hyperbolic plane, which are tilings by tiles of infinite size, can be wrapped over a TPMS to represent vertices and edges of an array in three-dimensional Euclidean space. In doing this, we harness the simplicity of a two-dimensional surface as compared with 3D space to build our catalogue. We numerically tighten these entangled flexible knits and nets to an ideal conformation that minimises the ratio of edge (or filament) length to diameter. To enable the tightening of periodic entanglements which may contain vertices, we extend the Shrink-On-No-Overlaps algorithm, a simple and fast algorithm for tightening finite knots and links. The ideal geometry of 3-periodic weavings found through the tightening process exposes an interesting physical property: Dilatancy. The cooperative straightening of the filaments with a fixed diameter induces an expansion of the material accompanied with an increase in the free volume of the material. Further, we predict a dilatant rod packing as the structure of the keratin matrix in the corneocytes of mammalian skin, where the dilatant property of the matrix allows the skin to maintain structural integrity while experiencing a large expansion during the uptake of water.