ScalingStacks

The results here hold for the ordinary Khovanov–Rozansky $\mathfrak{gl}_{N}$ link homologies as well as for their $\mathrm{GL(N)}$-equivariant and deformed versions [Lee05, Kho06, BNM06, Wu12, ETW18]. In the case of links in $S^{3}=\partial B^{4}$, the passage from the ordinary to deformed settings gives rise to spectral sequences that were studied in [Gor04, Ras15, Wu09, RW16]. Lobb and Wu [Lob09, Wu09], following pioneering work of Rasmussen [Ras10], showed that the associated filtrations for the generically deformed knot homologies in $S^{3}=\partial B^{4}$ contain lower bounds on the slice genus, i.e. the minimal genus of smooth surfaces in $B^{4}$ bounding the knot. Using such invariants, Freedman–Gompf–Morrison–Walker have outlined a strategy for testing counterexamples to the smooth 4-dimensional Poincaré conjecture [FGMW10]. One motivation for studying 4-manifold invariants from Khovanov–Rozansky homologies is that analogous spectral sequences might give rise to lower bounds on the genera of smooth surfaces in 4-manifolds $W^{4}$ bounding knots in $M^{3}=\partial W^{4}$.