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Khovanov skein lasagna modules with $1$-dimensional inputs

Qiuyu Ren, Ian Sullivan, Paul Wedrich, Michael Willis, Melissa Zhang

TL;DR

This work constructs a 1-dimensional-input variant of Khovanov skein lasagna modules, $ar{ abla}_0^2$, by integrating Rozansky–Willis homology for links in connected sums of $S^1 imes S^2$ and establishing functoriality for cobordisms in a broad class of 4-manifolds called 4-dimensional relative 1-handlebody complements. The authors develop a two-pronged approach: first, they prove functoriality for Rozansky–Willis homology in concrete olinebreak[4] settings using bypasses from Sullivan–Zhang, and second, they turn cobordisms inside-out to realize a dual theory via $ ilde{KhR}_2^+$. Key technical innovations include a detailed analysis of the diffeomorphism group of $D^2 imes S^2$ rel boundary, the introduction of Gluck twists to handle non-spin inputs, and a careful treatment of Rozansky projectors and belt-like configurations. The results yield structure theorems, invariance under Gluck twists, and a robust functorial framework for 4-manifold invariants derived from categorified link homology, enabling algorithmic computation on 1-handlebodies and connecting to Rozansky–Willis theory on more complex 4-manifolds. This advances the TQFT-like program for skein lasagna modules by providing a tractable, dimensionally constrained input category and a functorial backbone for higher-dimensional cobordism maps. The work also establishes foundational results about spin and non-spin settings, with implications for understanding exotic 4-manifolds and their interaction with categorified link homologies.

Abstract

We construct a variant of Khovanov skein lasagna modules, which takes the Khovanov homology in connected sums of $S^1\times S^2$ defined by Rozansky and Willis as the input link homology. To carry out the construction, we prove functoriality of Rozansky-Willis's homology for cobordisms in a class of $4$-manifolds that we call $4$-dimensional relative $1$-handlebody complements, by using, as a bypass, an isomorphism proved by Sullivan--Zhang relating the Rozansky-Willis homology and the classical Khovanov skein lasagna module of links on the boundary of $D^2\times S^2$. Along the way, we also present new results on diffeomorphism groups, on Gluck twists for Khovanov skein lasagna modules, and on the functoriality of $\mathfrak{gl}_2$ foams.

Khovanov skein lasagna modules with $1$-dimensional inputs

TL;DR

This work constructs a 1-dimensional-input variant of Khovanov skein lasagna modules, , by integrating Rozansky–Willis homology for links in connected sums of and establishing functoriality for cobordisms in a broad class of 4-manifolds called 4-dimensional relative 1-handlebody complements. The authors develop a two-pronged approach: first, they prove functoriality for Rozansky–Willis homology in concrete olinebreak[4] settings using bypasses from Sullivan–Zhang, and second, they turn cobordisms inside-out to realize a dual theory via . Key technical innovations include a detailed analysis of the diffeomorphism group of rel boundary, the introduction of Gluck twists to handle non-spin inputs, and a careful treatment of Rozansky projectors and belt-like configurations. The results yield structure theorems, invariance under Gluck twists, and a robust functorial framework for 4-manifold invariants derived from categorified link homology, enabling algorithmic computation on 1-handlebodies and connecting to Rozansky–Willis theory on more complex 4-manifolds. This advances the TQFT-like program for skein lasagna modules by providing a tractable, dimensionally constrained input category and a functorial backbone for higher-dimensional cobordism maps. The work also establishes foundational results about spin and non-spin settings, with implications for understanding exotic 4-manifolds and their interaction with categorified link homologies.

Abstract

We construct a variant of Khovanov skein lasagna modules, which takes the Khovanov homology in connected sums of defined by Rozansky and Willis as the input link homology. To carry out the construction, we prove functoriality of Rozansky-Willis's homology for cobordisms in a class of -manifolds that we call -dimensional relative -handlebody complements, by using, as a bypass, an isomorphism proved by Sullivan--Zhang relating the Rozansky-Willis homology and the classical Khovanov skein lasagna module of links on the boundary of . Along the way, we also present new results on diffeomorphism groups, on Gluck twists for Khovanov skein lasagna modules, and on the functoriality of foams.

Paper Structure

This paper contains 57 sections, 42 theorems, 76 equations, 67 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Khovanov skein lasagna modules
  4. Khovanov skein lasagna modules of D2*S2 and I*S1*S2
  5. Rozansky-Willis homology for framed links
  6. Properties of the Rozansky projector
  7. A lasagna interpretation of Rozansky-Willis homology
  8. Outline
  9. The input homology
  10. Definition of Khovanov skein lasagna modules with 1-dimensional inputs
  11. Turning cobordisms inside out
  12. Organization of the remaining sections
  13. Functoriality in concrete #S1*S2
  14. The statements
  15. Decomposition into elementary cobordisms
  16. ...and 42 more sections

Key Result

Theorem 1.1

There is a well-defined invariant $\bar{\mathcal{S}}_0^2(X;L)$ for pairs $(X,L)$ of compact oriented smooth $4$-manifold $X$ and framed oriented link $L\subset\partial X$, which is a $\mathbb Q$-vector space graded by $(\tfrac{1}{2}\mathbb Z)^2\times H_2(X,L;\mathbb Z/2)$. When $X=\natural^n(S^1\tim

Figures (67)

  • Figure 1: Left: Diagram of an admissible link near a surgery region. Right: A Rozansky projector.
  • Figure 2: The family of maps that constructs $P_{4,0}^\vee$ via the cobar construction in hogancamp2020constructing (grading shifts suppressed).
  • Figure 3: The Kirby diagram for the concrete $\#_{i=1}^3\natural^{m_i}(D^2\times S^2)$ with $3$-handles drawn, $(m_1,m_2,m_3)=(3,0,1)$. The $4$-handle is the one touching the outer belt point of each $3$-handle.
  • Figure 4: The belt-slide isomorphism \ref{['eq:SZ_slide']}, shown near one surgery region.
  • Figure 5: Type \ref{['item:concrete_finger']}--\ref{['item:concrete_slide']} moves in Proposition \ref{['prop:concrete_decomposition']}. Here, for (iv), we have only drawn, for simplicity, the case of pushing a crossing between the first two strands, but it is understood that the similar pushing is allowed for crossings between any two adjacent strands.
  • ...and 62 more figures

Theorems & Definitions (85)

  • Theorem 1.1: Definition \ref{['def:S02_bar']}
  • Theorem 1.2: Theorem \ref{['thm:concrete_functoriality']}
  • Theorem 1.3: Theorem \ref{['thm:diff_D2S2_rel_boundary']}
  • Corollary 1.4
  • Theorem 1.5: Theorem \ref{['thm:gluck']}
  • Theorem 1.6: Theorem \ref{['thm:gl2_webs_functorial']}
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • ...and 75 more