Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

     

On the cycling operation in braid groups

The cycling operation is a special kind of conjugation that can be applied to elements in Artin’s braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the cycling problem as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type, endowed with the Artin Garside structure.On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USSs), using left normal forms of braids. But one can equally use right normal forms and compute right-USSs. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USSs. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type.     

On the singular braid monoid of an orientable surface

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.

     

The cyclic sliding operation in Garside groups

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are rigid (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.     

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, υ ∈ B _n and an automorphism φ ∈ Aut(B _n ), decides whether υ = (φ(x))^?1 ux for some x ∈ B _n . As a corollary, we deduce that each group of the form B _n ? H, a semidirect product of the braid group B _n by a torsion-free hyperbolic group H, has solvable conjugacy problem.