Stabilization in the braid groups I: MTWS

Abstract

Choose any oriented link type $\mathcal{X}$ and closed braid representatives $X_{+},X_{-}$ of $\mathcal{X}$, where $X_{-}$ has minimal braid index among all closed braid representatives of $\mathcal{X}$. 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 $X_{+}^{\prime },X_{-}^{\prime }$) there is a sequence of closed braid representatives $X_{+}^{\prime }=X^1\to X^2\to \cdots \to X^r=X_{-}^{\prime }$ such that each passage $X^i\to X^{i+1}$ is strictly complexity reducing and non-increasing on braid index. The templates which define the passages $X^i\to X^{i+1}$ include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index $m\ge 4$ a finite set $\mathcal{T}\left(m\right)$ of new ones. The number of templates in $\mathcal{T}\left(m\right)$ is a non-decreasing function of $m$. We give examples of members of $\mathcal{T}\left(m\right),m\ge 4$, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.