The Markov Theorem for transverse knots

TL;DR

This paper proves a version of the Markov Theorem for transverse knots, establishing that isotopic transverse knots can be represented by isotopic transverse braids, using adapted proof techniques and new machinery for special cases.

Contribution

It extends the Markov Theorem to the setting of transverse knots, providing a new proof framework and addressing special cases with novel methods.

Findings

Established the Markov Theorem for transverse braids.

Developed new machinery for preferred longitudes in transverse setting.

Connected techniques from previous work and arc-presentations.

Abstract

A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are isotopic as transverse knots are also isotopic as transverse braids. The methods of the proof are based on Birman and Menasco's proof of the Markov Theorem in their recent paper (BM02), modified to the transverse setting. The modification is straightforward until we get to the special case of preferred longitudes, where we need some new machinery. We use techniques from earlier work by the author with Birman (BW00), by Birman and Menasco ((BM4), for example), and develop new methods from Cromwell's paper on arc-presentations (Cr95).