Related papers: Small braids having a big Ultra Summit Set
A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the…
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…
Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We…
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups;…
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure…
We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups…
The unknotting number $u$ and the genus $g$ of braid positive knots are equal, as shown by Rudolph. We prove the stronger statement that any positive braid diagram of a genus $g$ knot contains $g$ crossings, such that changing them produces…
It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an…
In this paper, we shall introduce two monoids. One is called a PM-monoid which contains the symmetric group, the other is called a braid PM-monoid which contains the braid group. We shall develop the theory of PM-monoids and that of braid…
For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful…
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…
Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen…
We estimate from above the set of knots, $\Omega(n,\mu)$, generated by closure of n-string 1+1- and 2+1-dimensional braids of irreducible length $\mu$ ($\mu>>1$) in the limit n>>1.
We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to…
We study the relationship between the number of full twists in positive braid representations of satellite links and their companion links. We construct infinitely many satellite links that admit positive braid representations with…
Upper bounds to the size of a family of subsets of an n-element set that avoids certain configurations are proved. These forbidden configurations can be described by inclusion patterns and some sets having the same size. Our results are…
Several distinct Garside monoids having torus knot groups as groups of fractions are known. For $n,m\geq 2$ two coprime integers, we introduce a new Garside monoid $\mathcal{M}(n,m)$ having as Garside group the $(n,m)$-torus knot group,…
In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension…
Let M to be a matroid defined on a finite set E. A subset L of E is locked in M if L is 2-connected in M, the complement of L is 2-connected in the dual M*, and min{r(L), r*(complement of L)} is greater than 1. In this paper, we prove that…
We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most…