Stabilization in the braid groups I: MTWS

Choose any oriented link type χ and closed braid representatives X ₊, X_- of X_-, where X_- has minimal braid index among all closed braid representatives of χ. 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 → ⋯ → X^r = 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 → X ⁱ⁺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 [6].