Studying links via closed braids. (English) Zbl 0835.57002

Bae, S. H. (ed.) et al., Lecture notes of the ninth KAIST mathematics workshop, Taejon, Korea, August 1-13, 1994. Vol. 1. Taejon: Korea Advanced Institute of Science and Technology, Mathematics Research Center, 1-67 (1994).
These lecture notes form a brisk introduction to links, considered as closures of braids. The author hopes that they may stimulate work towards an algorithm for deciding whether two links are isotopic. The standard existence and uniqueness theorems of Alexander and Markov have recently been given new constructive proofs and several of these are discussed here. Two solutions for the conjugacy problem for the braid group \(B_n\) are presented, due to Garside and to Xu. (The latter applies only to \(B_3\), at present). The algebraic structure of the braid groups is not otherwise considered. The latter half of these notes study the singular foliations induced on spanning surfaces for a link by a braid presentation for the link, and use related geometric ideas to re-examine Markov’s theorem.
For the entire collection see [Zbl 0824.00012].


57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups