62 research outputs found
On reduction curves and Garside properties of braids
In this paper we study the reduction curves of a braid, and how they can be
used to decompose the braid into simpler ones in a precise way, which does not
correspond exactly to the decomposition given by Thurston theory. Then we study
how a cyclic sliding (which is a particular kind of conjugation) affects the
normal form of a braid with respect to the normal forms of its components.
Finally, using the above methods, we provide the example of a family of braids
whose sets of sliding circuits (hence ultra summit sets) have exponential size
with respect to the number of strands and also with respect to the canonical
length.Comment: 20 pages, 3 figure
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 {\it
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.
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-USS), using left normal forms of braids. But one can equally use right
normal forms and compute right-USS's. Hard instances of the conjugacy search
problem correspond to elements having big (left and right) USS's. 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.Comment: 20 page
Conjugacy problem for braid groups and Garside groups
We present a new algorithm to solve the conjugacy problem in Artin braid
groups, which is faster than the one presented by Birman, Ko and Lee. This
algorithm can be applied not only to braid groups, but to all Garside groups
(which include finite type Artin groups and torus knot groups among others).Comment: New version, with substantial modifications. 21 pages, 2 figure
On the structure of the centralizer of a braid
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 elements if n=2k, and at most
$k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, due to work
of N.V.Ivanov and of S.J.Lee. Finally, we describe how one can explicitly
compute this generating set.Comment: Section 5.3 is rewritten. The proposed generating set is shown not to
be minimal, even though it is the smallest one reflecting the geometric
approach. Proper credit is given to the work of other researchers, notably to
N.V.Ivano
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