New quantum obstructions to sliceness

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 sn for each n ≥ 2, and where we have s 2 = 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 as well as to new sliceness obstructions that are not equivalent to concordance homomorphisms.2010 Mathematics Subject Classification. 57M25.

Upsilon-like concordance invariants from 𝔰𝔩n knot cohomology

“…The second example -the s invariant (or more precisely −s/2) -was due to Rasmussen [Ras10] and had a purely combinatorial definition in terms of the Lee perturbation [Lee02] of Khovanov cohomology. The reason for our normalization convention in Definition 1.1 is that there is a slew of such invariants (see for example [Wu09], [Lob09], [LL16]) arising from sl n Khovanov-Rozansky cohomology, the original definition of which [KR08] assigns negative quantum gradings to positive knots.…”

“…Whether this independence can be pushed so far that one can find a knot K for which ‫ג‬ n (K) is non-zero for some n and all other known sliceness obstructions vanish, is perhaps less interesting than finding some new topological applications of ‫ג‬ and related quantum invariants. It is not obvious, for example, that ‫ג‬ is insensitive to torsion elements of the concordance group (this is also non-obvious for the unreduced concordance invariants given in [LL16]).…”

“…In previous work [LL16], the authors studied the associated graded vector space to the cohomology Gr j H i ∂w (D) := F j H i ∂w (D) F j−1 H i ∂w (D) in the cases when ∂w is a product of distinct linear factors. In these cases, the bigraded complex vector space Gr j H i ∂w (D) is an invariant of K, it is of total dimension n, and supported in degree i = 0.…”

“…The quantum gradings of the support of the cohomology give rise to lower bounds on the smooth 4-ball genus of the knot, and it was a principal object of [LL16] to demonstrate that these bounds are heavily dependent on the choice of ∂w, and display interesting behavior from various points of view.…”

A Bennequin-Type Inequality and Combinatorial Bounds

“…The family of the slice torus invariants, as defined above, include (a suitable normalization of) the τ invariant from knot Floer homology [20], as well as sl n generalizations of the s-invariants [15]. In [12] T. Kawamura proved that in (KwC15) the s-invariant can be replaced with any slice-torus invariant.…”