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Reducible braids and Garside Theory

Juan González-Meneses and Bert Wiest

Algebraic & Geometric Topology 11 (2011) 2971–3010

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.

braid group, Garside group, Nielsen–Thurston classification, algorithm
Mathematical Subject Classification 2010
Primary: 20F10, 20F36
Received: 10 May 2011
Accepted: 28 June 2011
Published: 25 November 2011
Juan González-Meneses
Departamento de Álgebra
Facultad de Matemáticas
Universidad de Sevilla
Apdo 1160
41080 Sevilla
Bert Wiest
UFR Mathématiques (UMR 6625 du CNRS)
Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex