On a Refinement of the Non-Orientable $4$-genus of Torus Knots

TL;DR

This paper refines the understanding of the non-orientable 4-genus of torus knots by analyzing Batson's surfaces, showing they minimize within certain classes, and characterizing possible invariants for these surfaces.

Contribution

It proves Batson's surfaces minimize the non-orientable 4-genus among surfaces with the same normal Euler number and characterizes the pairs of invariants for these surfaces.

Findings

Batson's surfaces minimize the non-orientable 4-genus among surfaces with the same normal Euler number.

Complete characterization of possible pairs of normal Euler number and first Betti number for certain torus knots.

Identification of classes of torus knots where Batson's surfaces are genus minimizers.

Abstract

In formulating a non-orientable analogue of the Milnor Conjecture on the $4$-genus of torus knots, Batson developed an elegant construction that produces a smooth non-orientable spanning surface in $B^4$ for a given torus knot in $S^3$. While Lobb showed that Batson's surfaces do not always minimize the non-orientable $4$-genus, we prove that they always do minimize among surfaces that share their normal Euler number. We also completely determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson's surfaces are non-orientable $4$-genus minimizers.