The Non-Orientable Four-Ball Genus of a New Infinite Family of Torus Knots

TL;DR

This paper computes the non-orientable 4-ball genus for a new infinite family of torus knots, providing counterexamples to Batson's conjecture through a combination of band surgeries and bounds.

Contribution

It introduces a novel method to determine the non-orientable 4-ball genus for an infinite family of torus knots, extending previous results and disproving Batson's conjecture.

Findings

Computed $eta_4$ for specific torus knots explicitly.

Generalized the result to a broader family of torus knots.

Provided counterexamples to Batson's conjecture.

Abstract

We extend previous work by using a combination of band surgeries and known bounds to compute $\gamma_4(T_{4n, (2n\pm1)^2 + 4n-2}) = 2n-1$ for all $n \geq 1$. We further generalize this result by showing that $\gamma_4(T_{4n + 2k, n(4n + 2k) - 1}) = \gamma_4(T_{4n + 2k, (n+2)(4n + 2k) - 1}) = 2n-1 + k$ for all $n \geq 1$ and $k \geq 0$. All knots in this family are counterexamples to Batson's conjecture.