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Reducible braids and
Garside Theory
Juan González-Meneses and Bert Wiest |
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Algebraic & Geometric Topology 11 (2011) 2971–3010
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 20F10, 20F36
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References |
Publication
Received: 10 May 2011
Accepted: 28 June 2011
Published: 25 November 2011
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Authors |
| Juan González-Meneses | |
| Departamento de Álgebra Facultad de Matemáticas IMUS Universidad de Sevilla Apdo 1160 41080 Sevilla Spain |
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| http://personal.us.es/meneses/ | |
| Bert Wiest | |
| UFR Mathématiques (UMR 6625 du
CNRS) Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France |
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| http://perso.univ-rennes1.fr/bertold.wiest | |
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