Abstract
Let X be a closed indefinite 4-manifold with \(b_+(X) = 3 \; (\textrm{mod} \; 4)\) 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.
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Acknowledgements
We thank Hokuto Konno for comments on a draft of this paper.
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