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Stabilization in the braid groups. I: MTWS. (English) Zbl 1128.57003

Summary: Markov’s well-known theorem relates any two closed braid representatives \(X_+, X_-\) of the same knot or link. The moves are conjugation, stabilization and destabilization, where the former (resp. latter) adds \(+1\) (resp. \(-1\)) to the braid index. While the moves are very simple, the chain of moves connecting two representatives is completely unpredictable. In this paper we prove a modified version of Markov’s Theorem that has additional structure: Assume that \(X_-\) has minimum braid index. We define a lexicographically ordered complexity measure on a closed braid representative of a given link type, the first entry being braid index, and prove there is a finite set of moves from \(X_+\) to a modification of \(X_-\), which is strictly complexity-reducing. The finite set of moves depends on the braid index of \(X_+\), but is otherwise independent of link type.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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