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.
- Pub Date:
- January 2015
- Mathematics - Geometric Topology;
- Mathematics - Quantum Algebra;
- 35 pages, 6 figures, 3 tables. Second version: minor changes, added Question 4.8 and elaborated on the last remark in the section Outlook. This version corresponds to the published article