Stabilization in the braid groups I: MTWS

Birman, Joan S¹ ; Menasco, William W²

¹ Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York NY 10027, USA
² Department of Mathematics, University at Buffalo, Buffalo NY 14260, USA

Geometry & topology, Tome 10 (2006) no. 1, pp. 413-540.

Résumé

Choose any oriented link type $\mathcal{X}$ and closed braid representatives $X_{+},X_{-}$ of $\mathcal{X}$, where $X_{-}$ has minimal braid index among all closed braid representatives of $\mathcal{X}$. 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 $\mathcal{T}\left(m\right)$ of new ones. The number of templates in $\mathcal{T}\left(m\right)$ is a non-decreasing function of $m$. We give examples of members of $\mathcal{T}\left(m\right),m\ge 4$, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

Détail

DOI : 10.2140/gt.2006.10.413

Keywords: knot, links, braids, stabilization, Markov's theorem, braid foliations, flypes, exchange moves

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