Mutual braiding and the band presentation of braid groups

TL;DR

This paper characterizes when a closed braid and its axis are mutually braided by translating geometric conditions into combinatorial braid word relations, providing a method to decide mutual braiding.

Contribution

It introduces a combinatorial approach using band generators to determine mutual braiding of closed braids and their axes, linking geometric and algebraic braid properties.

Findings

Develops a criterion for mutual braiding based on band relations

Provides a combinatorial method to decide mutual braiding

Connects geometric conditions with braid word manipulations

Abstract

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close control on the way in which the whole family of fibre surfaces meet the family of discs spanning the braid axis. We show how such a braid can be presented naturally as a word in the `band generators' of the braid group discussed by Birman, Ko and Lee in their recent account of the band presentation of the braid groups. In this context we are able to convert the conditions for mutual braiding into the existence of a suitable sequence of band relations and other moves on the braid word, and derive a combinatorial method for deciding whether a braid is mutually braided.