Geometry & Topology
- Geom. Topol.
- Volume 10, Number 1 (2006), 413-540.
Stabilization in the braid groups I: MTWS
Choose any oriented link type and closed braid representatives of , where has minimal braid index among all closed braid representatives of . 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 and which replace them with closed braids ) there is a sequence of closed braid representatives such that each passage is strictly complexity reducing and non-increasing on braid index. The templates which define the passages include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index a finite set of new ones. The number of templates in is a non-decreasing function of . We give examples of members of , but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.
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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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. https://projecteuclid.org/euclid.gt/1513799713