Correction terms and the non-orientable slice genus

TL;DR

This paper develops a new lower bound for the non-orientable slice genus of knots using negative surgeries, signature, and concordance invariants, improving previous bounds and revealing superadditivity properties.

Contribution

It introduces a strengthened lower bound for the non-orientable slice genus based on knot invariants, extending prior results and highlighting superadditivity effects.

Findings

New lower bound for non-orientable slice genus using negative surgeries.

Bound coincides with existing bounds for specific classes of knots.

Superadditivity of the bound on stable non-orientable genus.

Abstract

By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus $\gamma_4(K)$ in terms of the signature $\sigma(K)$ and the concordance invariants $V_i(\overline{K})$, which strengthens a previous bound given by Batson, and which coincides with Ozsv\'ath-Stipsicz-Szab\'o's bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable non-orientable genus is sometimes better than the one on $\gamma_4(K)$.