Braids: a survey. (English) Zbl 1094.57006

Birman, Joan S.; Brendle, Tara E.

Menasco, William (ed.) et al., Handbook of knot theory. Amsterdam: Elsevier (ISBN 0-444-51452-X/hbk). 19-103 (2005).

This survey is about Artin’s Braid group \(B_n\) and its role in knot theory. The first goals of this paper are to provide enough of the essential background to make this text accessible to graduate students and to present those parts of the subject in which major progress was made or interesting new proofs of known results were discovered. The authors define the braid group and pure braid group in three distinct ways (they give a proof that two of them yield the same group and references to the literature establish the isomorphism in the remaining case). Examples are given which show how braids play a role in parts of mathematics that seem far away from knot theory. The authors explore the foundations of the close relationship between knots and braids. Proofs are given of the theorem of Alexander and of the theorem first formulated by A. Markov (which gives “moves” relating any two closed braid representatives of a knot or link, while simultaneously preserving the closed braid structure).
New results which relate to the study of knots via closed braids use the theory of braid foliations of a Seifert surface bounded by a knot which is represented as a closed braid. Three applications of braid foliations are given. The authors show ways in which structures first discovered in the braid groups generalize to structures in Garside groups, Artin groups and surface mapping class groups. Open problems are noted. A guide to computer software is given together with an extensive bibliography. Changing knots and links to closed braids, Garside’s algorithm for the word and conjugacy problems, and the Nielsen-Thurston classification of mapping classes in M\(_{0,n+1}\) are also presented.
For the entire collection see [Zbl 1073.57001].

Reviewer: Corina Mohorianu (Iaşi)

MSC:

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

Keywords:

braid groups; Artin groups; knots; links; braid foliations; representations