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arXiv · 1008.0238

Reducible braids and Garside theory

Juan Gonzalez-Meneses·Bert Wiest
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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.

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Juan Gonzalez-Meneses, Bert Wiest. 2010-08-02. Reducible braids and Garside theory. https://doi.org/10.2140/agt.2011.11.2971

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