Conjugacy in Garside Groups III: Periodic braids

Joan S. Birman; Volker Gebhardt; Juan Gonzalez-Meneses

Home/Archive/arXiv

arXiv · math/0609616

Conjugacy in Garside Groups III: Periodic braids

Joan S. Birman·Volker Gebhardt·Juan Gonzalez-Meneses

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.GT math.GR

Keep this discovery

BibTeX RIS

Joan S. Birman, Volker Gebhardt, Juan Gonzalez-Meneses. 2007-02-22. Conjugacy in Garside Groups III: Periodic braids. https://arxiv.org/abs/math/0609616

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗

arXiv2026

A counterexample to the wrapping number conjecture

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

math.GT↗

arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗