Untwisting 3-strand torus knots

TL;DR

This paper proves the sharpness of the signature bound for the topological 4-genus of 3-strand torus knots, and refines bounds for knots with more strands, advancing understanding of their genus properties.

Contribution

It establishes the exactness of the signature bound for 3-strand torus knots and improves bounds for 4- and 6-strand cases, using McCoy's twisting method.

Findings

Signature bound is sharp for 3-strand torus knots.

Bound is off by at most 1 for 4- and 6-strand torus knots.

Upper bound on the asymptotic ratio between topological 4-genus and Seifert genus improved from 2/3 to 14/27.

Abstract

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus of torus knots from 2/3 to 14/27.