Slice genus bound in $DTS^2$ from $s$-invariant

TL;DR

This paper proves that the $s$-invariant provides a lower bound on the slice genus of knots in the real projective 3-space, extending to more general cases with partial proofs.

Contribution

It confirms a conjecture relating the $s$-invariant to slice genus in $ ext{RP}^3$ and proposes a broader conjecture with partial validation.

Findings

$s$-invariant bounds null-homologous slice genus in $ ext{RP}^3$

Partial evidence for a more general slice genus bound

Extension of the $s$-invariant's applicability to non-null-homologous cases

Abstract

We prove a recent conjecture of Manolescu-Willis which states that the $s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lower bound on its null-homologous slice genus in the unit disk bundle of $TS^2$. We also conjecture a lower bound in the more general case where the slice surface is not necessarily null-homologous, and give its proof in some special cases.