Studying links via closed braids. III. Classifying links which are closed 3-braids

Joan Birman and William W. Menasco

Vol. 161 (1993), No. 1, 25–113

DOI: 10.2140/pjm.1993.161.25

Abstract

A complete solution is given to the classification problem for oriented links which are closed three-braids. The Classification Theorem asserts that, up to a finite list of exceptional cases, links which can be represented by closed 3-braids are represented by a unique conjugacy class in the group of 3-braids. The exceptional cases are the expected ones (links of braid index 1 and 2) and an unexpected infinite family of invertible links, each member of which has two 3-braid axes. The two axes correspond to diagrams which are related by “braid-preserving flypes”.

An algorithm is given which begins with an arbitrary closed 3-braid (or alternatively any link diagram with 3 Seifert circles), and converts it into a normal form which characterizes its oriented link type in oriented 3-space. One can decide from the normal form whether the link is prime or composite, split or irreducible, amphicheiral and or invertible. One can decide if the braid index is 3, 2 or 1. Using related results of P. J. Xu, one may determine the genus and construct a surface of maximum Euler characteristic with boundary the given link.

It is proved that the stabilization index of a link which is represented by a closed 3-braid is ≤ 1, i.e. any two 3-braid representatives of the same link type become conjugate after a single stabilization to B₄.

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 15 November 1990

Revised: 15 July 1992

Published: 1 November 1993