Stabilization distance bounds from link Floer homology

TL;DR

This paper introduces new invariants derived from link Floer homology to measure the complexity of surfaces bounded by knots in the 4-ball, providing bounds on stabilization and double point distances and constructing examples with large distances.

Contribution

It defines novel Floer homology-based invariants for pairs of surfaces in the 4-ball and demonstrates their effectiveness in bounding stabilization and double point distances, including explicit computations.

Findings

Invariants give lower bounds on stabilization and double point distances.

Computed invariants for specific pairs of slice disks using a trace formula.

Demonstrated existence of slice disks with arbitrarily large distances.

Abstract

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consider a double point distance between two surfaces of the same genus, which is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice disk analogue of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.