Related papers: Knots, Braids, and Knot Invariants

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon…

The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed…

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…

We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a…

We calculate Jones polynomials $V(H_r,t)$ for a family of alternating knots and links $H_r$ with arbitrarily many crossings $r$, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ for the associated graphs $G_+(H_r)$ and evaluating these…

We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem…

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical…

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…

In this expositional essay, we introduce some elements of the study of groups by analysing the braid pattern on a knitted blanket. We determine that the blanket features pure braids with a minimal number of crossings. Moreover, we determine…

We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the M\"obius function. We also prove a conjecture about further divisibility…

Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly…

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology,…

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot…

2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known…

A knot invariant is called skein if it is determined by a finite number of skein relations. In the paper we discuss some basic properties of skein invariants and mention some known examples of skein invariants.

Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined…

This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose…

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials…

These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and…