In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if… (More)
This paper is the second in a series (the others are  and ) in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set U SS(X) of an element X in a Garside group G is a finite set of elements in G, introduced in , which is a complete invariant of the conjugacy… (More)
We prove a conjecture due to Makanin: if α and β are elements of the Artin braid group B n such that α k = β k for some nonzero integer k , then α and β are conjugate. The proof involves the Nielsen-Thurston classification of braids.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in , are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups . We also present… (More)
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… (More)
An element in Artin's braid group B n is said to be periodic if some power of it lies in the center of B n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known… (More)
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most k(k+1) 2… (More)
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.
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 Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface. Vassiliev knot invariants were introduced by V. A. Vassiliev ([V1], [V2]; see also [B2], [B-N1]), and they have… (More)