Transverse Homology and Transverse Nonsimplicity

Abstract

We show that if a transversely nonsimple knot has two braid representatives, related by a negative flype, that can be distinguished by augmentation numbers of transverse homology, then an infinite family of transversely nonsimple knots with the same property can be constructed explicitly. As an application, we give an example of an infinite family of knots that can be proven to be transversely nonsimple by transverse homology, but not by the theta-hat invariant from knot Floer homology. In addition, we exhibit an infinite family of transversely nonsimple knots all of whose mirror images are transversely nonsimple as well; moreover, we prove that if a knot with the aforementioned property is also thin, then the theta-hat invariant must be trivial for all transverse representatives of the knot or its mirror image. Finally, a question posed by Plamenevskaya is as follows: does there exist a right-veering link for which the theta-hat invariant vanishes? A weaker version of this question asks if there exists a nondestabilizable right-veering braid for which the theta-hat invariant vanishes. We answer the latter in the affirmative by providing an example.