Relative Khovanov-Jacobsson classes

TL;DR

This paper introduces a new invariant derived from Khovanov homology for surfaces in the 4-ball, which can obstruct sliceness, detect specific slice knots, and is unaffected by certain connected sums.

Contribution

It defines a relative Khovanov-Jacobsson class invariant for surfaces in the 4-ball, extending previous invariants and providing new tools for knot and surface analysis.

Findings

Can obstruct sliceness of knots

Detects a pair of slices for 9_{46}

Unaffected by connected sums with knotted 2-spheres

Abstract

To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.