A note on transversal knots which are closed 3-braids

The topological classification of knots that are closed 3-braids is shown to lead to a classification theorem for tranversal knots that are represented by closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids. A transversal knot type is said to be transversally simple if it is determined, up to transversal isotopy, by its topological knot type and its Bennequin or self-linking invariant. In the absence of other invariants, it was natural to ask was whether all transversal knot types are transversally simple. In the manuscript [3] the authors of this note proved that the answer is no, by exhibiting an infinite family of pairs of transversal knot types, all closed 3-braids, which are smoothly isotopic, but not transversally isotopic. Soon after [3] was posted, additional examples of the same type were found by Etnyre and Honda [4]. The proofs in both [3] and [4] were indirect, and did not lead to computable invariants. In the years since [3] was posted there was new work, aimed at finding computable invariants of transversal knot type. In [10] Olga Plamenevskaya studied knots via their 2-fold branched covers, hoping that techniques from Heegaard Floer Homology on the covering spaces would detect the