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

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 3braid 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.