A counterexample to Batson's conjecture

TL;DR

This paper presents a counterexample to Batson's conjecture by demonstrating that the torus knot T_{4,9} bounds a smooth M"obius band in the 4-ball, challenging previous assumptions about non-orientable slice genera.

Contribution

The authors provide the first known counterexample to Batson's non-orientable Milnor conjecture using explicit construction.

Findings

T_{4,9} bounds a smooth M"obius band in the 4-ball

Counterexample disproves Batson's conjecture on non-orientable slice genus

Advances understanding of non-orientable surfaces in 4-dimensional topology

Abstract

We show that the torus knot $T_{4,9}$ bounds a smooth M\"obius band in the $4$-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.

Full text

Batson’s conjecture says that the smooth non-orientable $4$-ball genus of a torus knot is realized by a simple construction. This is analogous to Milnor’s conjecture (verified by Kronheimer-Mrowka [3]) that the smooth orientable $4$-ball genus of a torus knot is realized by the surface obtained from applying Seifert’s algorithm to a standard diagram of the knot.

The conjecture.

Let $T_{p,q}\subset S^{3}$ be the $(p,q)$ torus knot for $p>q\geq 2$, and let $D_{p,q}$ be the usual $q$-stranded braid closure diagram of $T_{p,q}$. Adding a blackboard-framed $1$-handle to two adjacent strands of $D_{p,q}$ results in a simpler torus knot, whose usual braid closure diagram we then consider. Repeating this procedure eventually arrives at the unknot, which may be capped off in the $4$-ball to give a surface $F_{p,q}\subset B^{4}$ with $\partial F_{p,q}=T_{p,q}$. Batson conjectured [1] that $b_{1}(F_{p,q})$ is minimal among the first Betti numbers of non-orientable smooth surfaces in the $4$-ball with boundary $T_{p,q}$.

We heard of this conjecture in a talk by Van Cott who, together with Jabuka, has verified it in many cases [2].

A counterexample.