Related papers: Thin position in the theory of classical knots

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$…

The aim of this note is to share the observation that the set of elementary operations of Turing on lattice knots can be reduced to just one type of simple local switches.

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,\xi)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact…

We study the bar-and-joint frameworks in $\mathbb{R}^2$ such that some vertices are constrained to lie on some lines. The generic rigidity of such frameworks is characterised by Streinu and Theran (2010). Katoh and Tanigawa (2013) remarked…

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use of techniques…

This is an article on the interaction between topology and physics which will appear in 1998 in a book called: A History of Topology, edited by Ioan James and published by Elsevier-North Holland.

We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact…

Thin elastic sheets appear in systems ranging from graphene to biological membranes, where phenomena such as wrinkling, folding, and thermal fluctuations originate from geometric nonlinearities. These effects are treated within weakly…

A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…

This paper deals with the interplay of the geometry of the norm and the weak topology in Banach spaces. Both dual and intrinsic connections between weak forms of rotundity and smoothness ared discussed. Weakly exposed points, weakly locally…

The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) We estimate the probability of a trivial knot formation on the lattice…

In this paper, we generalize the \textit{Clock Theorem} of Formal Knot Theory to knotoids in $S^2$. The clock theorem implies that clock states of a knotoid diagram form a lattice under transpositions. These states form the basis of many…

This paper provides a brief overview for non-specialists of some of the highlights in the development of the theory of weak interactions during the past century.

In this chapter, we present a review of latent position models for networks. We review the recent literature in this area and illustrate the basic aspects and properties of this modeling framework. Through several illustrative examples we…

This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods…

We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the…

Several complementary approaches to investigate knotted solutions of Maxwell's equations in vacuum are now available in literature. However, only partial results towards a unified description of them have been achieved. This is potentially…

In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized…

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads…