Reducible braids and Garside theory

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen-Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation.Comment: 28 pages, 4 figure

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem

On the centralizer of generic braids

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Presentations for the monoids of singular braids on closed surfaces

We give presentations, in terms of generators and relations, for the monoids SBn(M) 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 Birman for the monoids of Singular Artin braids.Dirección General de Enseñanza Superior e Investigación CientíficaTraining and Mobility of Researchers Programm

Improving an algorithm to solve multiple simultaneous conjugacy problems in braid groups

There are recent cryptographic protocols that are based on Multiple Simultaneous Conjugacy Problems in braid groups. We improve an algorithm, due to Sang Jin Lee and Eonkyung Lee, to solve these problems, by applying a method developed by the author and Nuno Franco, originally intended to solve the Conjugacy Search Problem in braid groups

Ordering pure braid groups on compact, connected surfaces

We prove that the pure braid groups on compact, connected, orientable surfaces are bi-orderable, and that the pure braid groups on compact, connected non-orientable surfaces have generalized torsion, thus they are not bi-orderable.Dirección General de Enseñanza Superior e Investigación Científic