Mutual braiding and the band presentation of braid groups

Details Mutual braiding and the band presentation of braid groups Mutual braiding and the band presentation of braid groups

1999-07-02

arxiv.org/abs/math/9907017v1

Knots in Hellas 98, Proceedings of the International Conference on Knot Theory and its Ramifications, ed. C.Gordon et al., Wor

Mathematics

Geometric Topology

13 pages, 10 figures. To appear in the proceedings of 'Knots in Hellas 98'

Scientific paper

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.

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