Geometry & Topology 10 (2006) 413–540

DOI: 10.2140/gt.2006.10.413

arXiv: math.GT/0310279

Abstract

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₊' = X¹→X²→…→X^r = X_-' such that each passage Xⁱ→Xⁱ⁺¹ is strictly complexity reducing and non-increasing on braid index. The templates which define the passages Xⁱ→Xⁱ⁺¹ include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index m≥ 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≥ 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