Studying links via closed braids. IV: Composite links and split links. (English) Zbl 0711.57006

Invent. Math. 102, No. 1, 115-139 (1990); erratum ibid. 160, No. 2, 447-452 (2005).
This is number 4 in a series of papers which study presentation of links as closed braids. The general aim is to find a succession of simple moves between closed braid representatives of the same link, such that the number of strings at each stage is no larger than the maximum for the two braids.
Here the goal is realized in the case of split or composite links, when one of the braid representatives has a simple form as an obvious split or composite braid. The extra move, which is introduced to avoid string- increasing applications of the Markov move, is called here an ‘exchange move’. It leaves the number of strings in the braid unchanged.
The main theorems, proved by the same manipulative techniques of the whole series of papers, are that any closed braid representing a composite or split link can be altered by exchange moves and conjugacies only to be a composite or split link. The major corollary is that (braid index-1) is additive under composition of knots.
Reviewer: H.Morton


57M25 Knots and links in the \(3\)-sphere (MSC2010)
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