New Quantum Obstructions to Sliceness

TL;DR

This paper explores new obstructions to sliceness in knot theory arising from perturbations of Khovanov-Rozansky cohomology, revealing that higher n invariants do not behave as previously expected and introduce novel sliceness obstructions.

Contribution

It demonstrates that for n >= 3, the invariants s_n do not arise generically and that perturbations produce new sliceness obstructions beyond known concordance homomorphisms.

Findings

s_n invariants for n >= 3 are not generic

Perturbations yield new sliceness obstructions

New concordance homomorphisms are introduced

Abstract

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 s_n for each n >= 2, and where we have s_2 = s. We demonstrate that s_n for n >= 3 does not in fact arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms as well as to new sliceness obstructions that are not equivalent to concordance homomorphisms.