Reducible braids and Garside Theory

Abstract

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. 20F10, 20F36

Fast Nielsen–Thurston classification of braids

“…Finally we recall the two following results from [27]: Theorem 5.16). Let x ∈ B n be a non-periodic, reducible braid which is rigid.…”

“…In [27], González-Meneses and Wiest showed that every element of (some refined version of) the set of Sliding Circuits of a reducible braid preserves a family of round or almost-round curves.…”

“…Conjecture 11. ( [27], Conjecture 3.5) Let x ∈ B n , with canonical length r. Let t be the minimal positive integer such that s k (x) = s t (x) for some k with 0 k < t. Then t is bounded by a polynomial in r and n.…”

“…Secondly, following the ideas introduced by Benardete, Gutierrez and Nitecki [1,2], connections between the reducibility problem and the Garside structures of braid groups have been used for detecting reducibility [1,2,28,27,12,10]. Our work fits in this approach and builds mainly on the last algorithm by González-Meneses and Wiest [27]. However Garside tools are not the only ones needed in the paper: we bring into play a very deep property of Mapping Class Groups: the so-called linearly bounded conjugator property [29,31], see Theorem 16.…”

How to read the length of a braid from its curve diagram

“…This paper is a branch of the paper [10], and I am extremely grateful to Juan González-Meneses for many helpful discussions.…”