Related papers: Complexity of relations in the braid group

We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $\sigma_1,\dots,\sigma_{n-1}$. Our study addresses…

We give a solution to the word problem for the singular braid monoid SB_n. The complexity of the algorithm is quadratic in the product of the word length and the number of the singular generators in the word. Furthermore we algebraically…

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or…

We suggest a new algorithm for finding a canonical representative of a given braid, and also for the harder problem of finding a $\sigma_1$-consistent representative. We conjecture that the algorithm is quadratic-time. We present numerical…

In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(\sigma_1^m)$ the normal closure of $\sigma_1^m$, where $\sigma_1$ is the classical generator of $B_n$. We characterize the conditions under…

This paper is devoted to the proof of a structural theorem, concerning certain homomorphic images of Artin braid group on $n$ strands in finite symmetric groups. It is shown that any one of these permutation groups is an extension of the…

A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all…

We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of…

E. Artin described all irreducible representations of the braid group B_k to the symmetric group S(k). We strengthen some of his results and, moreover, exhibit a complete picture of homomorphisms of B_k to S(n) for n<2k+1. We show that the…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

We describe a new algorithm which for each braid returns a quasi-geodesic sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears either only positively or only negatively.

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group $\mathcal{S}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in \mathcal{S}_n$ are $\sigma$-equivalent, if there are natural numbers $k$…

We define pseudo-Garside groups and prove a theorem about them parallel to Garside's result on the word problem for the usual braid groups. The main novelty is that the set of simple elements can be infinite. We introduce a group B=B(Z^n)…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

We classify an action of the $n$-strand braid group on the free group of rank $n$ which is similar to the Artin representation in the sense that the $i$-th generator $\sigma_{i}$ of $B_{n}$ acts so that it fixes all free generators $x_{j}$…

The language of maximal lexicographic representatives of elements in the positive braid monoid $A_n$ with $n$ generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and…

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or…

In this note we study a family of algebras with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between…

A new presentation of the $n$-string braid group $B_n$ is studied. Using it, a new solution to the word problem in $B_n$ is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes…

Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n, as the quotient of the braid group B_n by the relations \sigma_i\sigma_{i+1}^{-1}\sigma_i=\sigma…