Recognizing destabilization, exchange moves and flypes

TL;DR

This paper presents an algorithmic method to recognize when a closed braid admits destabilization, exchange moves, or flypes, key isotopies in braid theory, simplifying the process of moving between braid representations without stabilization.

Contribution

It provides a novel algorithmic solution to the recognition problem for key isotopies in the MTWS calculus, enabling efficient identification of these moves.

Findings

Algorithm successfully recognizes destabilization, exchange moves, and flypes.

Complexity measure can be monotonically simplified.

Enhances understanding of braid isotopies and their recognition.

Abstract

The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed---destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"---determining when a given closed $n$-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for these three key isotopies of the MTWS calculus. The algorithm is "directed" by a complexity measure that can be {\em monotonically simplified} by the application of "elementary moves".