Properties of closed 3-braids

Details Properties of closed 3-braids Properties of closed 3-braids

2006-06-19

arxiv.org/abs/math/0606435v2

Mathematics

Geometric Topology

68 pages, 10 figures, 3 tables; rev 10 Oct 07: some reorganization; added sections 4.3, 7.1, 7.7

Scientific paper

We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid words with Morton-Williams-Franks bound 3 and show that closed positive braids of braid index 3 are closed positive 3-braids. For closed braids on more strings, we study the alternating links occurring. In particular we classify those of braid index 4, and show that their Morton-Williams-Franks inequality is exact. Finally, we use the Burau representation to obtain new braid index criteria, including an efficient 4-braid test.

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