Twisted braids

TL;DR

This paper extends classical braid theory to twisted braids within twisted knot theory, proving a twisted Alexander theorem, establishing Markov moves, and providing simplified group presentations.

Contribution

It introduces a twisted Alexander theorem and simplified group presentations for twisted braid groups, generalizing classical braid results to twisted knot theory.

Findings

Any twisted link can be represented as the closure of a twisted braid.

The twisted braid group has a reduced presentation with minimal generators.

The paper establishes Markov-type moves for twisted braids.

Abstract

Twisted knot theory, introduced by M.O. Bourgoin, is a generalization of virtual knot theory. It naturally yields the notion of a twisted braid, which is closely related to the notion of a virtual braid due to Kauffman. In this paper, we first prove that any twisted link can be described as the closure of a twisted braid, which is unique up to certain basic moves. This is the analogue of the Alexander Theorem and the Markov Theorem for classical braids and links. Then we also give reduced presentations for the twisted braid group and the flat twisted braid group. These reduced presentations are based on the fact that these twisted braid groups on $n$ strands are generated by a single braiding element and a single bar element plus the generators of the symmetric group on $n$ letters.