On the slice genus of quasipositive knots in indefinite 4-manifolds

TL;DR

This paper establishes a new lower bound on the genus of surfaces in indefinite 4-manifolds bounded by quasipositive knots, linking it to the slice genus and extending adjunction inequalities.

Contribution

It introduces a novel genus bound for surfaces in 4-manifolds with specific properties, generalizing adjunction inequalities and relating to quasipositive knots.

Findings

Lower bound matches slice genus for null-homologous cases

Bound differs from minimal genus by at most 1 in symplectic cases

Extended adjunction inequality to classes of negative self-intersection

Abstract

Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 \; ({\rm mod} \; 4)$ and with non-vanishing mod $2$ Seiberg--Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in $X \setminus B^4$ representing a given homology class and with boundary a quasipositive knot $K \subset S^3$. In the null-homologous case our inequality implies that the minimal genus of such a surface is equal to the slice genus of $K$. If $X$ is symplectic then our lower bound differs from the minimal genus by at most $1$ for any homology class that can be represented by a symplectic surface. Along the way, we also prove an extension of the adjunction inequality for closed $4$-manifolds to classes of negative self-intersection without requiring $X$ to be of simple type.