On the values taken by slice torus invariants

TL;DR

This paper characterizes the possible values of slice-torus invariants on knots, linking them to the stable smooth slice genus, and shows that the local Thom conjecture resolution implies the existence of such invariants without explicit constructions.

Contribution

It provides a characterization of slice-torus invariants' values in terms of the stable smooth slice genus, connecting knot invariants to a major conjecture.

Findings

Values of slice-torus invariants are characterized by the stable smooth slice genus.

Resolution of the local Thom conjecture implies existence of slice-torus invariants.

No explicit knot homology constructions needed for the invariants.

Abstract

We study the space of slice-torus invariants. In particular we characterize the set of values that slice-torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory.