Entanglement complexity of lattice ribbons

We consider a discrete ribbon model for double-stranded polymers where the ribbon is constrained to lie in a three-dimensional lattice. The ribbon can be open or closed, and closed ribbons can be orientable or nonorientable. We prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes, and a theorem about the frequency of occurrence of certain patterns in these ribbons. We use this to derive results about the frequency of knots in closed ribbons, the linking of the boundary curves of orientable closed ribbons, and the twist and writhe of ribbons. We show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite-n limit. For an orientable ribbon, the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n, and similar results hold for the twist and writhe.     

The writhe of permutations and random framed knots

We introduce and study the writhe of a permutation, a circular variant of the well‐known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non‐Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017     

A note on regular isotopy of singular links

We show that two isotopic oriented 4-valent singular link diagrams with transverse intersections are regularly isotopic if and only if they have the same writhe and the same rotation number.

     

Subspaces of knot spaces

The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.

     

Geometry and Mechanics of Uniform n-Plies: from Engineering Ropes to Biological Filaments

We study the mechanics of uniform n-plies, correcting and extending previous work in the literature. An n-ply is the structure formed when n pretwisted strands coil around one another in helical fashion. Such structures are encountered widely in engineering (mooring ropes, power lines) and biology (DNA, proteins). We first show that the well-known lock-up phenomenon for n=2, described by a pitchfork bifurcation, gets unfolded for higher n. Geometrically, n-plies with n>2 are all found to behave qualitatively the same. Next, using elastic rod theory, we consider the mechanics of n-plies, allowing for axial end forces and end moments while ignoring friction. An exact expression for the interstrand pressure force is derived, which is used to investigate the onset of strand separation in plied structures. After defining suitable displacements we also give an alternative variational formulation and derive (nonlinear) constitutive relationships for torsion and extension (including their coupling) of the overall ply. For a realistic loading problem in which the ends are not free to rotate one needs to consider the topological conservation law, and we show how the concepts of link and writhe can be extended to n-plies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Helical and Localised Buckling in Twisted Rods: A Unified Analysis of the Symmetric Case

We review the geometric rod theory for the case of a naturallystraight, linearly elastic, inextensible, circular rod suffering bendingand torsion but no shear. Our primary focus is on the post-bucklingbehaviour of such rods when subjected to end moment and tension.Although this is a classic problem with an extensive literature, datingback to Kirchhoff, the usual approach tends to neglect the physicalinterpretation of solutions (i.e., rod configurations) to the modelsproposed. Here, we explicitly compute geometrical properties of buckledrods. In a unified approach, making use of Kirchhoff's dynamic analogy,both the classical helical and the more recently investigated localisedbuckling are considered. Special attention is given to a consistenttreatment of concepts of link, twist and writhe.