Related papers: Braids for pretzel links
The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.
The goal of this paper is to discuss the possibility of finding an algorithm that can give all distinct knots up to a desired complexity. Two such algorithms are presented, one based on projections on a plane, the other on closed…
In the last decade, a number of public key cryptosystems based on com- binatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts…
We present a consistent definition for braided ribbon networks in 3-dimensional manifolds, unifying both three and four valent networks in a single framework. We present evolution moves for these networks which are dual to the Pachner moves…
We give an alternate expansion of the colored Jones polynomial of pretzel links which recovers the degree formula in arXiv:1807.00957. As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle…
It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals…
In this paper we study the theory of {\it pseudo knots}, which are knots with some missing crossing information, and we introduce and study the theory of {\it pseudo tied links} and the theory of {\it pseudo knotoids}. In particular, we…
We consider the greatest common divisor (GCD) of all sums of $k$ consecutive terms of a sequence $(S_n)_{n\geq 0}$ where the terms $S_n$ come from exactly one of following six well-known sequences' terms: Pell $P_n$, associated Pell $Q_n$,…
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.
A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between…
We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by…
We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.
Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure…
We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals…
We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.
Extended welded links are a generalization of Fenn, Rim\'{a}nyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of…
We summarize ideas and methods that apply to rigidity and connectivity percolation. These include: constraint counting concepts, exact calculations on diluted trees and numerical results on diluted lattices using matching algorithms. The…
Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…
We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently…
We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well…