Homological width and Turaev genus

Abstract

Khovanov homology and knot Floer homology are generalizations of the Jones polynomial and the Alexander polynomial respectively. They are bigraded Z-modules, and their underlying polynomials are recovered by taking the graded Euler characteristic. The two homologies share many characteristics, however their relationship has yet to be fully understood. In both Khovanov homology and knot Floer homology, the two gradings can be combined into a single diagonal grading. Homological width is a measure of the support of the homology with respect to the diagonal grading. In this thesis, we show that the homological width of Khovanov homology and knot Floer homology have a common upper bound. Every link diagram has an associated Turaev surface, a certain Heegaard surface in the 3-sphere on which the knot has an alternating projection. The Turaev genus of a knot is the minimum genus of a Turaev surface where the minimum is taken over all diagrams of the knot. Turaev introduced this surface in order to prove a conjecture about the span of the Jones polynomial. Previously, it has been shown that Turaev genus gives an upper bound for the homological width of Khovanov homology. Since Khovanov homology is a generalization of the Jones polynomial, one might expect that Turaev genus and Khovanov homology are related. In this thesis, we show that Turaev genus also gives an upper bound for the homological width of knot Floer homology, giving the first known relationship between the Alexander polynomial and the Turaev surface. It is also more evidence towards a relationship between Khovanov homology and knot Floer homology. In addition, we construct infinite families of links whose Khovanov homology have the same homological width. Using this construction, we compute the Khovanov width of all closed 3-braids.