Algebraic & Geometric Topology 11 (2011) 2971–3010

DOI: 10.2140/agt.2011.11.2971

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.

Keywords

braid group, Garside group, Nielsen–Thurston classification, algorithm

Mathematical Subject Classification

Primary: 20F10, 20F36

References

Publication

Received: 10 May 2011
Accepted: 28 June 2011
Published: 25 November 2011

Authors