Abstract
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.
Citation
Joan S Birman. William W Menasco. "Stabilization in the braid groups I: MTWS." Geom. Topol. 10 (1) 413 - 540, 2006. https://doi.org/10.2140/gt.2006.10.413
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