2 00 8 A note on closed 3 - braids ∗

This is a review article about knots and links of braid index 3. Its goal is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. As an example, non-invertibility is a property of knots which is fairly subtle, yet we know (see Theorem 2 below) precisely which links of braid index 3 are and are not invertible. Therefore 3-braids could be very useful for testing potential invariants that might detect non-invertibility. The purpose of this note is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. It includes the work in an earlier preprint [4] which was restricted to 3-braid representatives of transversal knots. The preprint [4] was written when [26] was in preparation, and the authors of [26] asked us about low crossing examples to test their then-new invariants of transversal knots. 1 Three basic theorems In this section we present the basic lemmas and theorems about links that are closed three-braids. Initially, we will use the classical presentation for the braid group B3: < σ1, σ2;σ1σ2σ1 = σ2σ1σ2 > (1) this article is an expanded version of the preprint “A note on transversal knots which are closed 3-braids”, arXiv math/0703669. partially supported by the US National Science Foundation Grant DMS 0405586. partially supported by the US National Science Foundation Grant DMS 0306062