Narrow Results

Two links A, B (where A is the unknot) are exchangeable if B is a braid rel A, and also A is a (generalized) braid rel B. If moreover A, B are mutually braided in the sense of Rudolph, i.e. their fibres meet in a particularly nice way, the whole situation can be described by a finite collection of combinatorial data, that I call a film. In this paper is proven that a film can be associated to each pair of mutually braided links, and that from a film it is always possible to reconstruct (uniquely up to isotopy) the whole situation of the pair of fibrations of two mutually braided links. Using this new tool, it is proven that exchangeable links are in fact mutually braided.

This work is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph [7]. 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 [1] 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 thus derive a combinatorial method for deciding whether a braid is mutually braided.