Knot Floer homology of positive braids

Abstract

We compute the next-to-top term of the knot Floer homology of any link obtained as the closure of a positive braid, showing in particular that the rank is one for any prime knot in this family. As such knots are fibered, it follows that their monodromies are fixed-point free. We compare the set of positive braids with other classes of knots known to have this property. One such class consists of knots possessing "diagonal" grid diagrams. We provide an example of such a knot that is not a positive braid, providing an answer to a question of Vance and Kubota. We conclude with a number of problems and questions for future study naturally motivated by our theorem.

v2: Matthew Hedden added as an author. Complete overhaul of v1. Stronger version of the main theorem stated as Theorem 3.5, which streamlines the proof and makes it more self-contained. Example of a diagonal knot which is not a positive braid included as Proposition 1.4. Added a concluding section containing open problems and questions. 15 pages, 3 figures. v1: 12 pages, 6 figures