Link cobordisms and absolute gradings on link Floer homology

Ian Zemke

Published 2017-01-12Version 1

We give an alternate construction of absolute Maslov and Alexander gradings on link Floer homology, using a surgery presentation of the link complement. We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. We also show that our construction agrees with the original definition of the symmetrized Alexander grading defined by Ozsv\'{a}th and Szab\'{o}. As an application, we compute the link cobordism maps associated to a closed surface in $S^4$ with a simple dividing set. For simple dividing sets, the maps are determined by the genus of the surface. We show how the grading formula quickly recovers some known bounds on the concordance invariants $\tau(K)$ and $\Upsilon_K(t)$, and use it to prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4--manifolds. As a final application, we recover an adjunction relation for the ordinary Heegaard Floer cobordism maps as a consequence of our grading formula and properties of the link Floer cobordism maps. This gives a link Floer interpretation of adjunction relations and inequalities appearing in Seiberg--Witten and Heegaard Floer theories.