On a Nonorientable Analogue of the Milnor Conjecture

TL;DR

This paper investigates a nonorientable analogue of Milnor's Conjecture for the 4-genus of knots, proving it for many torus knots and deriving new formulas for their signatures.

Contribution

It proves Batson's conjecture for many torus knots using a lower bound on nonorientable 4-genus and provides new signature formulas for these knots.

Findings

Proved the conjecture for many infinite families of torus knots.

Derived new closed formulas for the signature of torus knots.

Established a connection between nonorientable 4-genus and knot signatures.

Abstract

The nonorientable 4-genus $\gamma_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $\gamma_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for $\gamma_4$ formulated by Ozsv\'ath, Stipsicz, and Szab\'o. As a side product we obtain new closed formulas for the signature of torus knots.