On the topological 4-genus of torus knots

TL;DR

This paper establishes an upper bound on the topological slice genus of large torus knots, showing it is less than three quarters of their ordinary genus, and provides a linear estimate based on the signature invariant.

Contribution

It introduces a new bound on the topological 4-genus of large torus knots and relates it to the signature invariant for the first time.

Findings

Topological 4-genus is less than 75% of the ordinary genus for large torus knots.

Derived the best linear estimate of the topological slice genus using the signature invariant.

Established a quantitative relationship between the topological 4-genus and knot invariants.

Abstract

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.