Related papers: Complexity of relations in the braid group

We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this…

From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $\rho: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation…

We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.

In the present paper, we construct a monomorphism from (Artin) pure braid group $PB_{n}$ into a group, which is `bigger' than $PB_{n}$. Roughly speaking, this mapping is defined on words of braids by adding `new generators' between…

Two words are $k$-binomially equivalent if each subword of length at most $k$ occurs the same number of times in both words. The $k$-binomial complexity of an infinite word is a counting function that maps $n$ to the number of $k$-binomial…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

Any positive word comprised of random sequence of tokens form a finite alphabet can be reduced (without change of length) using an appropriate size Braid group relationships. Surprisingly the Braid relations dramatically reduce the…

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer…

For $n\geq 2$, let $G_n$ be a group and let $\rho: B_n\rightarrow G_n$ be a representation of the braid group $B_n$. For a field $\mathbb{K}$ and $a,b,c\in \mathbb{K}$, Bardakov, Chbili, and Kozlovskaya extend the representation $\rho$ to a…

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary…

We generalize the Brin-Higman-Thompson groups $n G_{k,1}$ to monoids $n M_{k,1}$, for $n \ge 1$ and $k \ge 2$, by replacing bijections by partial functions. The monoid $n M_{k,1}$ has $n G_{k,1}$ as its group of units, and is…

In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard…

In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.

We show that the virtual singular braid monoid on $n$ strands embeds in a group $VSG_n$, which we call the virtual singular braid group on $n$ strands. The group $VSG_n$ contains a normal subgroup $VSPG_n$ of virtual singular pure braids.…

In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this…

A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found.…

Guralnick, Kantor, Kassabov and Lubotzky (J. Eur. Math. Soc. 13.2, 2011, 391-458) [GKKL] give 3-generator 7-relator presentations of $A_n$ and $S_n$ with bit-length $O(\log n)$ for $n\geq5$. This is the best possible bit-length, since…

We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed…

We show that the word problem of the Brin-Higman-Thompson group $n G_{k,1}$ is {\sf coNP}-complete for all $n \ge 2$ and all $k \ge 2$. For this we prove that $n G_{k,1}$ is finitely generated, and that $n G_{k,1}$ contains a subgroup of $2…

We study a specific line arrangement obtained from a generic $2$-section of the braid arrangement, and compute the fundamental group of its complement via braid monodromy. We show that the resulting presentation of the fundamental group…