Distinguishing slice disks using knot Floer homology

TL;DR

This paper introduces a knot Floer homology invariant to classify and distinguish slice disks of knots, providing new tools for understanding their isotopy classes and complexity.

Contribution

It defines a new invariant in knot Floer homology for classifying slice disks and demonstrates its effectiveness in distinguishing non-isotopic disks and analyzing their complexity.

Findings

Computed the invariant for slice disks obtained by deform-spinning.

Showed the invariant distinguishes non-isotopic slice disks with diffeomorphic complements.

Introduced a numerical stable diffeomorphism invariant called the rank.

Abstract

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a slice disk of a composite knot, we define a numerical stable diffeomorphism invariant called the rank. This can be used to show that a slice disk is not a boundary connected sum, and to give lower bounds on the complexity of certain hyperplane sections of the slice disk.