Ribbon homology cobordisms

Khovanov homology and cobordisms between split links

“…This follows from the injectivity results mentioned above ([17] for ribbon concordance, Theorem 1.4 for strongly homotopy–ribbon concordance) along with recent work by Lipshitz and Sarkar [18] showing that Khovanov homology detects split links. We also give a second proof using Heegaard Floer homology, making use of a similar injectivity result due to Daemi, Lidman, Vela‐Vick, and Wong [9]. This result provides another example of nondecreasing simplicity under ribbon concordance, as mentioned above.…”

“…They then used this result along with a careful analysis of the Ozsváth–Szabó spectral sequence from $$\widetilde{\operatorname{Kh}}(L)$$ to $$\widehat{\operatorname{HF}}(\Sigma (\overline{L}))$$ [22] to deduce the analogous detection result for Khovanov homology (both reduced and unreduced), as mentioned above. Now, if $$L_0$$ is ribbon concordant to $$L_1$$, then a result of Daemi, Lidman, Vela‐Vick, and Wong [9, Theorem 1.19] shows that $$\widehat{\operatorname{HF}}(\Sigma (L_0))$$ injects into $$\widehat{\operatorname{HF}}(\Sigma (L_1))$$ as a summand, and it is not hard to see that this holds at the level of $$\mathbb {F}[X]/(X^2)$$–modules as well. The proof of Theorem 1.5 then follows along the same lines as above.…”

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“…By [4, Proposition 2.1] the induced map 𝑅 𝑁 (𝑌 ) → 𝑅 𝑁 (𝑋) is surjective. Both of these results follow from a result of Gerstenhaber-Rothaus [5, Theorem 1(ii)] which allows one to extend a representation 𝜌 ∶ 𝜋 1 (𝑋) → 𝑆𝑂(𝑁) to a representation 𝜌 ′ ∶ 𝜋 1 (𝑌 ) → 𝑆𝑂(𝑁) which restricts to 𝜌 using the fact that 𝑌 has a handle decomposition with 𝑛 1-handles and 𝑛 2-handles added to a collar neighborhood of 𝑋, and so that the 2-handles homologically cancel the 1-handles to obtain a homology cobordism (this is called a ribbon homology cobordism in [4]). Note that the map 𝑅 𝑁 (𝑌 ) → 𝑅 𝑁 (𝑋) may be with respect to different coordinates, since the generators of 𝜋 1 (𝑋) may be regarded as a subset of the generators of 𝜋 1 (𝑌 ), and hence this polynomial map is a projection onto the subspace…”

“…Gordon answers this conjecture for knots satisfying various hypotheses, as a special case if 𝐾 0 or 𝐾 1 is fibered. Much more evidence has been amassed for this conjecture: if 𝐾 0 ≥ 𝐾 1 ≥ 𝐾 0 , then 𝐾 0 and 𝐾 1 have the same S-equivalence class [6, Theorem 1.6], Seifert genus and knot Floer homology [16,Theorem 1.4], Khovanov homology [11,Corollary 2], and instanton knot Floer homology [4,Corollary 4.5], [10,Theorem 7.4].…”

“…In the conclusion we point out that Theorem 1.2 potentially generalizes to homology ribbon cobordism in the sense of [4] and we consider the possibility of answering some other questions from [7, Section 6].…”