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


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 Xr = 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.

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