On the nonorientable four-ball genus of torus knots

TL;DR

This paper introduces a new lower bound for the nonorientable four-ball genus of knots using advanced homology theories, demonstrating sharpness for certain torus knots and proving non-existence of certain surfaces for others.

Contribution

It develops a novel lower bound on the nonorientable four-ball genus using involutive and unoriented knot Floer homology, with applications to specific torus knots.

Findings

Lower bound is sharp for families of torus knots including $T_{4n,(2n ext{±}1)^2}$.

Torus knots $T_{p,q}$ with even $p$ and non-square $p/2$ do not bound locally flat M"obius bands for almost all $q$ coprime to $p$.

Counterexamples to Batson's conjecture are identified among certain torus knots.

Abstract

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus $\gamma_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge 2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat M\"obius band for almost all integers $q$ relatively prime to $p$.