Reducible braids and Garside Theory

González-Meneses, Juan; Wiest, Bert

doi:10.2140/agt.2011.11.2971

González-Meneses, Juan ¹ ; Wiest, Bert ²

¹Departamento de Álgebra, Facultad de Matemáticas, IMUS, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain
²UFR Mathématiques (UMR 6625 du CNRS), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2971-3010

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Résumé

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.

DOI : 10.2140/agt.2011.11.2971

Classification : 20F10, 20F36
Keywords: braid group, Garside group, Nielsen–Thurston classification, algorithm

Affiliations des auteurs :

González-Meneses, Juan ¹ ; Wiest, Bert ²

¹ Departamento de Álgebra, Facultad de Matemáticas, IMUS, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain
² UFR Mathématiques (UMR 6625 du CNRS), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

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González-Meneses, Juan; Wiest, Bert. Reducible braids and Garside Theory. Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2971-3010. doi: 10.2140/agt.2011.11.2971

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