Volume 10, issue 1 (2006)

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Stabilization in the braid groups I: MTWS

Joan S Birman and William W Menasco

Geometry & Topology 10 (2006) 413–540
 arXiv: math.GT/0310279
Abstract

Choose any oriented link type $\mathsc{X}$ and closed braid representatives ${X}_{+},{X}_{-}$ of $\mathsc{X}$, where ${X}_{-}$ has minimal braid index among all closed braid representatives of $\mathsc{X}$. The main result of this paper is a ‘Markov theorem without stabilization’. It asserts that there is a complexity function and a finite set of ‘templates’ such that (possibly after initial complexity-reducing modifications in the choice of ${X}_{+}$ and ${X}_{-}$ which replace them with closed braids ${X}_{+}^{\prime },{X}_{-}^{\prime }$) there is a sequence of closed braid representatives ${X}_{+}^{\prime }={X}^{1}\to {X}^{2}\to \cdots \to {X}^{r}={X}_{-}^{\prime }$ such that each passage ${X}^{i}\to {X}^{i+1}$ is strictly complexity reducing and non-increasing on braid index. The templates which define the passages ${X}^{i}\to {X}^{i+1}$ include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index $m\ge 4$ a finite set $\mathsc{T}\left(m\right)$ of new ones. The number of templates in $\mathsc{T}\left(m\right)$ is a non-decreasing function of $m$. We give examples of members of $\mathsc{T}\left(m\right),m\ge 4$, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

Keywords
knot, links, braids, stabilization, Markov's theorem, braid foliations, flypes, exchange moves
Mathematical Subject Classification
Primary: 57M25, 57M50
Publication
Received: 23 June 2005
Accepted: 25 January 2006
Published: 27 April 2006
Proposed: Robion Kirby
Seconded: Benson Farb, David Gabai
Authors
 Joan S Birman Department of Mathematics Barnard College Columbia University 2990 Broadway New York NY 10027 USA William W Menasco Department of Mathematics University at Buffalo Buffalo NY 14260 USA