Stabilization in the braid groups I: MTWS

Choose any oriented link type $X$ and closed braid representatives $X_{+}, X_{-}$ of $X$ , where $X_{-}$ has minimal braid index among all closed braid representatives of $X$ . 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 $X_{+}^{'}, X_{-}^{'}$ ) there is a sequence of closed braid representatives $X_{+}^{'} = X^{1} \to X^{2} \to \dots \to X^{r} = X_{-}^{'}$ 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 $m \geq 4$ a finite set $T (m)$ of new ones. The number of templates in $T (m)$ is a non-decreasing function of $m$ . We give examples of members of $T (m), m \geq 4$ , 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

References

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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

Volume 10, issue 1 (2006)

Joan S Birman and William W Menasco

Abstract