A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds (1910.08195v2)
Abstract: We extend the definition of Khovanov-Lee homology to links in connected sums of $S1 \times S2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S1 \times S2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: $B2 \times S2$, $S1 \times B3$, $\mathbb{CP}2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.