Conjugacy in Garside Groups III: Periodic braids

Journal of Algebra (Impact Factor: 0.6). 10/2006; 316(2). DOI: 10.1016/j.jalgebra.2007.02.002
Source: arXiv


An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.

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    • "In the pseudo-Anosov case, they also give the structure of the unstable foliation F u , its measure and the dilatation factor λ. The dynamic of a braid only depends on its conjugacy class, in particular so does its Nielsen-Thurston type. So we can study the geometry of any given β ∈ B n , through any conjugate β = α −1 βα: Periodicity is easily recognizable in braid groups [5], and β k = ∆ 2s if and only if β k = ∆ 2s . If β is reducible, and C is a reduction system for β (1-manifold such that β(C) = C), then α(C) is a reduction system for β. "
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    ABSTRACT: In this work we present a natural surjective map from rigid braids in B_3 (in Garside sense) to SL_2(N). This map provides an upper and a lower bound for the dilatation factor of a pseudo-Anosov 3-strand braid. These bounds only depend on the canonical length of the classical Garside structure of B_3.
    Preview · Article · Jul 2013 · Journal of Knot Theory and Its Ramifications
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    • "If they are not the same, answer " x and y are not conjugate " and STOP. (2) If x and y are periodic use [4] and STOP. (3) If x and y are reducible, use [9] and STOP. "
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    ABSTRACT: We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
    Preview · Article · Apr 2012 · Journal of Group Theory
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    • "As a self-homeomorphism on n-punctured 2-disk, an n-braid β is of one of the following three dynamic types known as Nielsen-Thurston classification [19]: (1) β is periodic if some power of β is isotopic to some power of Dehn twist along the boundary of disk; (2) β is reducible if there exists a family of pairwise disjoint essential non-peripheral simple closed curves in D n , called reduction system, which is preserved by a n-braid isotopic to β; (3) β is pseudo-Anosov is neither (1) nor (2). The conjugacy problem for periodic braids is relatively easy [11] [4]. For reducible braids, once the reduction system is known, the conjugacy problem can be reduced into several pieces, which are for either periodic or pseudo-Anosov braids. "
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    ABSTRACT: We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge.
    Preview · Article · Mar 2012 · Journal of Knot Theory and Its Ramifications
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