Reducible braids and Garside theory. (English) Zbl 1252.20035

González-Meneses, Juan; Wiest, Bert

Algebr. Geom. Topol. 11, No. 5, 2971-3010 (2011).

Summary: 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.

MSC:

20F36	Braid groups; Artin groups
20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
68W30	Symbolic computation and algebraic computation
68Q25	Analysis of algorithms and problem complexity

Keywords:

braid groups; Garside groups; reducible braids; Nielsen-Thurston classification; polynomial time algorithms; polynomial time complexity; cyclic sliding operation; conjugacy classes

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