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2011 Reducible braids and Garside Theory
Juan González-Meneses, Bert Wiest
Algebr. Geom. Topol. 11(5): 2971-3010 (2011). DOI: 10.2140/agt.2011.11.2971


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 González-Meneses. Bert Wiest. "Reducible braids and Garside Theory." Algebr. Geom. Topol. 11 (5) 2971 - 3010, 2011.


Received: 10 May 2011; Accepted: 28 June 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1252.20035
MathSciNet: MR2869449
Digital Object Identifier: 10.2140/agt.2011.11.2971

Primary: 20F10 , 20F36

Keywords: algorithm , Braid group , Garside group , Nielsen–Thurston classification

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.11 • No. 5 • 2011
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