Geometry & Topology

Stabilization in the braid groups I: MTWS

Joan S Birman and William W Menasco

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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+=X1X2Xr=X such that each passage XiXi+1 is strictly complexity reducing and non-increasing on braid index. The templates which define the passages XiXi+1 include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index m4 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),m4, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

Article information

Geom. Topol., Volume 10, Number 1 (2006), 413-540.

Received: 23 June 2005
Accepted: 25 January 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds

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


Birman, Joan S; Menasco, William W. Stabilization in the braid groups I: MTWS. Geom. Topol. 10 (2006), no. 1, 413--540. doi:10.2140/gt.2006.10.413.

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