Relative Khovanov-Jacobsson classes

Abstract

To a smooth, compact, oriented, properly-embedded surface in the 4-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for 946; is not hindered by detecting connected sums with knotted 2-spheres.

Categorical lifting of the Jones polynomial: a survey

“…For closed surfaces, this invariant turns out not to be interesting: it vanishes if some component of the surface is not a torus, and otherwise is 2 n if the surface consists of n tori [146,60]. On the other hand, for surfaces with boundary a nontrivial link in S 3 , Khovanov homology does give an interesting invariant [169], even distinguishing some surfaces that are topologically isotopic [64].…”

Categorical lifting of the Jones polynomial: a survey

“…For closed surfaces, this invariant turns out not to be interesting: it vanishes if some component of the surface is not a torus, and otherwise is 2 n if the surface consists of n tori [60,145,170]. On the other hand, for surfaces with boundary a nontrivial link in S 3 , Khovanov homology does give an interesting invariant [168], even distinguishing some surfaces that are topologically isotopic [64,110].…”

Khovanov homology and exotic surfaces in the 4-ball

“…We can isolate B so that C#S decomposes into a link cobordism C (S \ D2 ) : L 0 → L 0 U followed by a saddle merging L 0 and U . By [SS21], the map induced by S \ D2 is identical to the map induced by the link cobordism induced by a standard D 2 in B .…”

“…An analogous argument applies to the reversed cobordisms, appealing instead to the surjectivity of the map Kh(L 2 ) → Kh(L 1 ) induced by the reverse of C . Remark 3.5 A similar (independently established) technique is used in [SS21] for finding prime knots with an arbitrarily large number of distinct (but non-exotic) slices. Moreover, a similar technique appears in [JZ20] for an invariant from [JM16] in knot Floer homology.…”

“…Connections with Khovanov-Jacobsson classes The approach we use was motivated by [SS21], where instead the surfaces are viewed as link cobordisms ∅ → K and the resulting induced maps are distinguished by their associated relative Khovanov-Jacobsson classes KJ Σ and KJ Σ , i.e., the classes to which they map the generator of Kh(∅) = Z, modulo sign. Although these classes are easily defined, they are generally impractical as an explicit obstruction, requiring significant computational endurance to both calculate and distinguish.…”