New quantum obstructions to sliceness

It is well known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in sl⁡(n) Khovanov–Rozansky cohomology, where a perturbation gives rise to the concordance homomorphisms sn for each n⩾2, and where we have s2=s. We demonstrate that sn for n⩾3 does not in fact arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms and to new sliceness obstructions that are not equivalent to concordance homomorphisms.