Stabilization In The Braid Groups

The main result, Theorem 1, is Markov's TheoremWithout Stabilization (MTWS) for links in 3-space. Choose any oriented link type X and any closed braid representatives X \Gamma ; X+ of X , where X+ has minimal braid index. The MTWS asserts that there is a complexity function anda finite set of `templates' suchthat (possibly after initial complexity-reducing modifications in the choices of X \Gamma ; X+ , which replace them with closed braids X + of the same braid index) there is a sequence of closed braid representatives X \Gamma =X =X such that each passage X is strictly complexity reducing and non-increasing on braid index. The templates which define the `moves' which take X include 3 familiar ones: the destabilization, exchange move and admissible flype templates, and in addition, for each braid index m 4 a finite set T (m) of templates. The number of templates in T (m) is an increasing function of m for sufficiently large m. We give examples of members of T (m), but not a complete listing