Link homology theories and ribbon concordances

Abstract

It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.2010 Mathematics Subject Classification. 57M27.

Split link detection for sl(P) link homology in characteristic P

“…By [Miy98], there is a ribbon concordance C b from K # to K b . There is an induced map KR N pC b ; Fq : KR N pK # ; Fq Ñ KR N pK b ; Fq which can be shown to be injective [Kan19,CGL `20] ultimately based on an arugment of Zemke for knot Floer homology [Zem19]. By the proof of [Wan20, Proposition 5.7], there are injective maps (displayed below as dotted) making the diagram KR N pD s , q; Fq KR N pK b , q; Fq KR N pL, q; Fq KR N pD # s , q; Fq KR N pK # , q; Fq KR N pL, q; Fq a a # KR N pC b ;Fq commute.…”

On Khovanov Homology and Related Invariants

“…This led to an exciting series of papers extending this result to various homology-type invariants for knots. Within the realm of Khovanov-type invariants, Levine-Zemke [28] extended the result to the original Khovanov homology, Kang [18] extended the result to a setup that includes Khovanov-Rozansky homologies [24], knot Floer homologies and other theories, and Sarkar [47] defined the notion of ribbon distance and derived bounds on this from Khovanov-Lee homology.…”

“…More generally, we show that a ribbon concordance between links induces injective maps on link homologies defined via webs and foams modulo relations. Kang provides a different approach in [18], where it is shown that a ribbon concordance induces injective maps on link homology theories that are multiplicative link TQFTs and which are either associative or Khovanov-like. Our proof relies mainly on the fact that all of the homology theories considered in Section 3 satisfy certain cutting neck and sphere relations in the category of dotted cobordisms, without the need to provide new definitions or develop special techniques.…”

Ribbon homology cobordisms

“…Sherry Gong has informed the authors of a direct proof of a version of Corollary 1.15 with coefficients in Z for concordances in Y − × I, without appealing to the isomorphism between KHI and I . Kang [Kan19] has very recently provided a general proof of Corollary 1.15 for conic strong Khovanov-Floer theories for concordances in S 3 × I, which may be used to recover a version of Corollary 1.15.…”