# Link cobordisms and absolute gradings on link Floer homology

## TL;DR

This paper establishes that link cobordism maps in link Floer homology are graded, derives a grading change formula, and applies these results to obtain bounds on knot invariants and determine cobordism maps from surface genus.

## Contribution

The paper introduces a grading change formula for link cobordism maps and demonstrates its applications to bounds on knot invariants and characterization of cobordism maps by surface genus.

## Key findings

- Proved that link cobordism maps are graded and satisfy a grading change formula.
- Derived a new bound for the Upsilon invariant of knots in negative definite 4-manifolds.
- Showed that cobordism maps for surfaces in S^4 are determined by surface genus.

## Abstract

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.