---
title: 'Skein Lasagna Modules: 4D Link Homology'
url: https://www.emergentmind.com/topics/skein-lasagna-modules
type: topic
---

# Skein Lasagna Modules: 4D Link Homology

Skein lasagna modules are 4-dimensional extensions of link homology in which properly embedded framed surfaces in a 4-manifold are combined with labeled “input balls,” and then quotiented by local replacement relations. In the Morrison–Walker–Wedrich framework, the skein lasagna module is the degree-zero part of the blob-homology package; when the ambient 4-manifold is \(B^4\), it canonically recovers the underlying link homology. Subsequent work has re-expressed the construction as a colimit over a skein category, a cabled handlebody formula, a homotopy colimit in a completed Bar–Natan setting, and several variants based on equivariant, deformed, Bar-Natan, Floer, Rozansky–Willis, and stable-homotopy inputs [2009.08520, 2206.04616, 2401.06600, 2602.13462].

## 1. Definition and formal structure

In a standard \(KhR_N\) formulation, a lasagna filling of \((W;L)\) consists of a finite collection of disjoint embedded 4-balls \(B_i\subset \operatorname{Int} W\), a properly embedded, framed, oriented surface \(\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i\) with \(\partial \Sigma=L\), and for each input ball a homogeneous label \(v_i\in KhR_N(B_i,L_i)\), where \(L_i=\Sigma\cap \partial B_i\). The degree-zero skein lasagna module is then obtained from the free abelian group on such fillings by imposing multilinearity in the labels and the ball-replacement relation: if a label \(v_i\) is realized by a lasagna filling inside a 4-ball, that filling may replace the ball carrying \(v_i\) without changing the class. In the blob-complex description, this is precisely \(S_0(W;L)=H_0(\Blob_\ast(W;L))\), and the elementary presentation is designed so that the degree-zero blob differential is built into the relations [2009.08520].

The grading conventions depend on the chosen input theory. In one \(KhR_N\) normalization, a filling \(F\) carries
\[
\deg(F)=\sum_{i=1}^r \deg(v_i)+(0,(1-N)\chi(\Sigma))\in \mathbb Z^2,
\]
while other formulations record additionally the relative homology class \([\Sigma]\in H_2(W;L)\), or treat the skein module as the blob-degree-zero piece of a triply graded theory [2009.08520, 2401.06600]. This is not a superficial bookkeeping issue: later applications to genus bounds, deformations, and stable refinements depend on exactly which gradings survive the quotient.

An alternative but equivalent viewpoint packages the construction as a colimit over a skein category. Here the objects are skeins \(\Sigma\subset X\) rel \(L\), the morphisms are embedded cobordisms between skeins, and a functor \(Z\) sends each input link to its link-homology module. The skein lasagna module is then
\[
\mathcal S^Z_0(X;L)=\operatorname{colim}\bigl(Z:\mathcal C(X;L)\to R\text{-Mod}\bigr),
\]
so concretely \((\Sigma,v)\sim (\Sigma',Z([S])v)\) whenever a morphism \(S:\Sigma\to\Sigma'\) is present [2602.13462]. In the earlier handle-decomposition work, this same structure is interpreted as the “Hilbert space” of a fully-extended \((4+\epsilon)\)-dimensional TQFT, with ball-replacement functioning as the local relation that makes gluing well defined [2206.04616].

## 2. Handle decompositions, cabled formulas, and homotopy colimits

For 4-manifolds built from \(B^4\) by 2-handle attachment, Manolescu–Neithalath give a concrete cabled description. If \(W\) is obtained by attaching \(n\) 2-handles along a framed link \(K=K_1\cup\cdots\cup K_n\subset S^3\), and \(\alpha\in \mathbb Z^n\cong H_2(W)\), they define a cabled Khovanov–Rozansky complex \(\mathcal C_\alpha(K)\) by summing over links \(K(r-\alpha^-,r+\alpha^+)\), then quotienting by the braid-group actions on the parallel strands and by dot-annihilation relations. The resulting theorem is a canonical isomorphism
\[
\Phi:\mathcal C_\alpha(K)\xrightarrow{\;\cong\;}S_0(W;\emptyset,\alpha).
\]
Over a field, the same paper proves a tensor-product formula for boundary-connected sums, and shows that adding 3- or 4-handles does not change \(S_0(W;\emptyset)\) [2009.08520].

The broader handle-decomposition formalism of Manolescu–Walker–Wedrich starts with a Morse filtration \(W_1\subset W_2\subset W_3\subset W_4\), where \(W_1\) is the 0- and 1-handlebody, \(W_2\) adds 2-handles, and \(W_3\) adds 3-handles. The 2-handle step is expressed by a direct sum over cabled skein modules of the 1-handlebody, modulo braid relations and ribbon cobordism relations \(\psi_i^{[d]}\); the 3-handle step is a coequalizer of the two hemisphere maps capping the equator of the attaching sphere; and 4-handles leave the module unchanged. This converts the computation of \(S(W;L)\) into an explicit sequence of cabling, quotienting, and coequalizing operations tied directly to a Kirby diagram [2206.04616].

A later reformulation replaces the ordinary colimit by a homotopy colimit. In the setting of \(\mathrm{KhR}_2\), the Manolescu–Neithalath cabled formula is reinterpreted as a mapping telescope in a completion of the category of complexes over Bar-Natan’s cobordism category. The directed system is assembled into a two-term double complex \(D\), and its totalization \(\operatorname{Tot}(D)\) is identified with the homotopy colimit \( \operatorname{hocolim}(A_k\to A_{k+1})\). Proposition 2.12 in that paper shows that this totalization satisfies the universal property of the colimit in the completed Karoubian/Bar-Natan category [2402.01081]. This makes precise why the 2-handlebody formula can be attacked with projector techniques and categorical telescopes rather than only with direct algebraic quotients.

By 2026, the gluing and handle-attachment picture had been generalized beyond \(\mathfrak{gl}_N\) theories. A uniform theory for any functorial link invariant \(H^\star\) in \(S^3\times I\) gives 1-, 2-, and 3-handle attachment formulas via a complete description of the gluing homomorphism and a cornered version of the skein module. In that setting the 2-handle step is governed by an “annular algebra” of cablings and cups/caps, and the 3-handle step becomes tensoring over \(H^\star(S^2)\) with the evaluation map on the trivial sphere [2602.17825]. This suggests that the handle-calculus aspect of skein lasagna theory is not specific to Khovanov–Rozansky inputs, but is a formal feature of functorial link theories.

## 3. Explicit computations, projectors, and finiteness phenomena

One of the earliest model computations is the disk bundle \(S^2\times D^2\), represented by a single 0-framed unknot. For general \(N\),
\[
S_0(S^2\times D^2;\emptyset)\cong \mathbb Z[A_1,\dots,A_{N-1},A_0^{\pm1}],
\]
with \(\deg_q(A_k)=-2k\) for \(k=0,\dots,N-1\), and the summand in relative class \(\alpha\in H_2(S^2\times D^2)\cong \mathbb Z\) is the subspace of homogeneous polynomials of total degree \(\alpha\). In the \(N=2\) case this yields a single copy of \(\mathbb Z\) in bidegrees \(j=-2k\) with \(\alpha=k\) and vanishing otherwise [2009.08520]. This computation became the base case for several later refinements.

A more specialized computation concerns \((S^2\times B^2,\tilde\beta)\), where \(\tilde\beta\subset S^1\times S^2\) is a geometrically essential boundary link. The key tool is the categorified Jones–Wenzl projector \(P_n\in \operatorname{Kom}(TL_n)\), defined as the filtered colimit \(P_n:=\operatorname{colim}_m FT_n^{\otimes m}\). It is idempotent up to homotopy, kills turnbacks, and “eats” braid generators. In the cabling telescope one inserts \(P_n\) on the core strands before capping off by the belt; Proposition 5.4 then shows that only the through-degree-0 Rozansky projector \(P^\vee_{n,0}\) survives when \(n\) is even, while the whole telescope is annihilated when \(n\) is odd. The resulting theorem is that
- if \(n\) is odd or \(\alpha\) is odd, then \(S_0^2(S^2\times B^2;\tilde 1_n,\alpha)=0\);
- if \(n=2k\) and \(\alpha\) is even, then
\[
S_0^2(S^2\times B^2;\tilde1_{2k},\alpha)\cong \mathbb F_{|\alpha|}[A_0,A_0^{-1},A_1]\otimes KhR_2(\operatorname{Tr}P^\vee_{2k,0}).
\]
The same paper identifies \(H^\ast(\operatorname{Tr}(\beta\otimes T_n^{\Omega_\alpha}))\) with \(S_0^2(S^2\times B^2;\tilde\beta,\alpha)\), and relates the projector term to the Rozansky–Willis invariant for nullhomologous links in \(S^2\times S^1\) [2402.01081].

The same projector analysis yields a vanishing theorem for \(S^2\times S^2\). Using the 0-framed Hopf link Kirby diagram, the module in class \((\alpha_1,\alpha_2)\) is viewed as an iterated colimit in the two cable directions. If either \(\alpha_1\) or \(\alpha_2\) is odd, the intermediate colimit already vanishes. When both are even, the dotted annulus map on \(H^\ast(\operatorname{Tr}P^\vee_{2k,0})\) is analyzed, and the relevant obstruction classes in \(\operatorname{Ext}(P^\vee_{2k,0},P^\vee_{2k+2,0})\) vanish or become nilpotent. The conclusion is
\[
S_0^2(S^2\times S^2;\emptyset,(\alpha_1,\alpha_2))=0
\]
for all levels, confirming a conjecture of Manolescu [2402.01081].

The theory is not uniformly finite-dimensional. For \(N=2\) over a perfect field \(k\), Manolescu–Walker–Wedrich compute
\[
\dim_k S(S^1\times B^3;S^1\times P_p)=
\begin{cases}
1,&p=0,\\
4,&p=1,\\
\infty,&p\ge 2,
\end{cases}
\]
and in fact the bidegree \((0,0)\) summand is already infinite-dimensional when \(p\ge2\) [2206.04616]. A common misconception is therefore that skein lasagna modules should behave like finite-rank TQFT state spaces in every local example; the explicit \(S^1\times B^3\) calculation shows that local infinite dimensionality is an intrinsic phenomenon of the theory.

## 4. Variants, deformations, and refinements

Several distinct inputs now support skein-lasagna-type constructions.

| Variant | Input theory | Characteristic result |
|---|---|---|
| Equivariant/deformed \(\mathfrak{gl}_N\) | \(KhR_{T(N)}, KhR, KhR_\Sigma\) | non-vanishing, genus bound, deformation decomposition |
| Bar-Natan | \(BN\) over \(\mathbb F_2[H]\) | dot-trading, neck-cutting, \(H\)-torsion gluing behavior |
| Floer lasagna | \(\widehat{HFL}\) | 2-handlebody model via cabled link Floer homology |
| 1-dimensional inputs | Rozansky–Willis/Khovanov in \(\#^n(S^1\times S^2)\) | handlebody model and lasso relation |
| Stable-homotopy refinement | Lipshitz–Sarkar spectra \(\mathcal X_{LS}\) | cohomology recovers \(\mathcal S^{Kh}_0\), but spectrum is stronger for \(L\neq\emptyset\) |

For equivariant and deformed \(\mathfrak{gl}_N\) theories, Morrison–Walker–Wedrich define \(S_H(W;L)\) using \(KhR_{T(N)}\), \(KhR\), or \(KhR_\Sigma\), with decomposition over \(H_2(W;L)\). In the \(SU(N)\)-equivariant theory, if \(S\subset W\) is smoothly embedded, oriented, and homologically diverse, then its lasagna class \([S]\in S_H(W;L)\) is non-torsion over \(R=\mathbb Z[e_1,\dots,e_N]\). They also define
\[
q_{\min}^N(\alpha)=\min\{q\text{-degree of a nonzero class in }S_H(W;L)/\operatorname{tors}\},
\]
and derive the bound
\[
\chi(S)\le \frac{-N\,\alpha\!\cdot\!\alpha-q_{\min}^N(\alpha)}{N-1}.
\]
In the deformed theory, a full decomposition theorem splits \(S_{KhR_\Sigma}(W;L)\) over colorings of the boundary-link components by the deformation parameters [2401.06600].

The Bar-Natan version replaces \(KhR_N\) by the Frobenius pair \(R^{BN}=\mathbb F_2[H]\), \(A^{BN}=R^{BN}[X]/(X^2-HX)\). Its local relations are birth/death of a small sphere, the dotted sphere relation, dot-trading, and neck-cutting. The resulting module \(S_0^{BN}(M;L)\) is an \(\mathbb F_2[H]\)-module, and connect-sum gluing maps preserve \(H\)-torsion order on a half-torsion-free submodule. The paper computes
\[
S_0^{BN}(S^4;\emptyset)\cong \mathbb F_2,\qquad
S_0^{BN}(\mathbb CP^2;\emptyset)\cong \mathbb F_2[H]\cdot\{[\mathbb CP^1]\},
\]
and
\[
S_0^{BN}(S^2\times S^2;\emptyset)\cong \mathbb F_2[H]\cdot\{[S^2\times pt],[pt\times S^2]\}
\]
[2504.03968].

Floer lasagna modules replace Khovanov–Rozansky homology by \(\widehat{HFL}\) and decorate the filling surface with dividing arcs cutting it into \(\Sigma_w\) and \(\Sigma_z\). The resulting module \(\mathcal{FL}(W,\mathbb L)\) is graded by the relative class in \(H_2(W,L)\), Maslov degree \(M(\mathcal F)=\chi(\Sigma_w)+\sum_i M(v_i)\), and Alexander degree \(A(\mathcal F)=\frac{\chi(\Sigma_w)-\chi(\Sigma_z)}2+\sum_i A(v_i)\). For 4-manifolds obtained by attaching 2-handles to \(B^4\), Chen proves a natural bigrading-preserving isomorphism between \(\mathcal{FL}(W,\mathbb L)\) and a cabled link Floer homology \(\widehat{cHFL}(\mathbb L;K)\) [2203.07650].

The 1-dimensional-input Khovanov theory allows input 1-handlebodies rather than only 4-balls, and uses Rozansky–Willis homology in connected sums of \(S^1\times S^2\). If \(X=\natural^n(S^1\times B^3)\), then
\[
\bar{\mathcal S}_0^2(X;L)\cong \widetilde{KhR}_2^-(L)
\]
canonically. A further handle-attachment theorem expresses the resulting module for a 1- and 2-handlebody as a coequalizer of two “lasso maps,” i.e. after quotienting cabled Rozansky–Willis homology by the lasso relation \(\Phi_i(v)=\Psi_i(v)\). This framework yields explicit disk-bundle computations, including
\[
\overline S^{2,O}_0(S^2\times D^2)\cong \mathbb Q[A_0,A_0^{-1}],\qquad
\overline S^{2,T}_0(S^2\times D^2)\cong \mathbb Q[A_0]/(A_0^2-1),
\]
as well as vanishing for \(D(p)\) when \(p>0\) and partial vanishing for \(\Sigma_g\times D^2\) in positive homological degree [2510.05273, 2606.29452].

A stable-homotopy refinement replaces the module-valued functor with the Lipshitz–Sarkar spectrum \(\mathcal X_{LS}\) and defines
\[
\mathcal E^{LS}_0(X;L)=\operatorname{hocolim}\bigl(\mathcal X_{LS}:\mathcal C(X;L)\to \mathbf\Sigma\bigr).
\]
Its reduced cohomology recovers the Khovanov skein lasagna module,
\[
\widetilde H^\ast(\mathcal E^{LS}_0(X;L))\cong \mathcal S^{Kh}_0(X;L),
\]
but for \(L\neq\emptyset\) the spectrum is stronger than the \(\mathfrak{gl}_2\) skein lasagna module because it retains higher operations such as Steenrod squares [2602.13462].

A further enrichment appears in the equivariant theory over \(R=\mathbb Z[E_1,\dots,E_N]\), where the module \(S_0^N(W,L;R)\) carries a well-defined action of \(U(\mathfrak{sl}_2)\). The construction uses green-dotted boundary data satisfying Euler-characteristic constraints, together with the infinitesimal \(\mathfrak{sl}_2\)-action on equivariant \(KR_N\)-homology and foam complexes [2604.02997].

## 5. Surface classes, genus bounds, and exotic smooth structures

A central point of skein lasagna theory is that a properly embedded oriented surface \(S\subset W\) with \(\partial S=L\) determines a canonical lasagna class. In the equivariant/deformed \(\mathfrak{gl}_N\) framework, this class depends only on the relative homology class \([S]\in H_2(W;L)\), and has tridegree
\[
\deg_t(S)=[S]\!\cdot\![S],\qquad
\deg_q(S)=(1-N)\chi(S)-N\,[S]\!\cdot\![S].
\]
The non-vanishing theorem for homologically diverse surfaces then makes the skein module a source of genus bounds. In particular, the quantity \(q_{\min}^N(W;L,[S])\) functions as a lasagna analogue of Rasmussen’s invariant and reproduces the classical Rasmussen bound when \(W=B^4\), \(N=2\), and \(L\) is a knot [2401.06600].

The \(\mathfrak{gl}_2\) theory admits a Lee deformation and a lasagna \(s\)-invariant. In that setting,
\[
\See(X;L)\cong \bigoplus_{(a^+,a^-)\in H_2(X,L)^2}\mathbb Q\langle x_{a^+,a^-}\rangle,
\]
and for \(a\in H_2(X)\),
\[
s(X;L;a)=q(x_{a,0}).
\]
This invariant satisfies normalization, symmetries, connected-sum and gluing properties, and an adjunction-type genus bound
\[
g(X;a)\ge \tfrac12\bigl(s(X;a)+a^2\bigr).
\]
The same paper applies the \(\mathfrak{gl}_2\) skein lasagna module to the exotic pair of knot traces \(X_{-1}(-5_2)\) and \(X_{-1}(P(3,-3,-8))\). In relative class \(1\), one finds
\[
\So_{0,q}(X_1;1)\otimes \mathbb Q=
\begin{cases}
\mathbb Q,& q=1,3,\\
0,&\text{otherwise,}
\end{cases}
\qquad
\So_{0,q}(X_2;1)\otimes \mathbb Q=
\begin{cases}
\mathbb Q,& q=\pm1,\\
0,&\text{otherwise.}
\end{cases}
\]
Hence \(\So(X_1)\not\cong \So(X_2)\), giving what the paper describes as the first analysis-free proof of the existence of exotic compact orientable 4-manifolds [2402.10452].

The Bar-Natan theory detects exotic surfaces with boundary rather than only closed-manifold phenomena. Hayden’s exotic pair \(F_g,F_g'\subset B^4\) determines a nonzero \(H\)-torsion difference class \(\delta_g^L=[F_g]-[F_g']\) of order at least \(2\) in \(S_0^{BN}(B^4;K_g)\). If \([S]\in S_0^{BN}(X;L')\) is primitive, Sullivan proves that after taking a connect sum with \(B^4\), the resulting difference class in \(S_0^{BN}(X\# B^4;L'\sqcup K_g)\) has the same \(H\)-torsion order as \(\delta_g\). The conclusion is that one internal stabilization is generally not enough for these exotic knotted surfaces [2504.03968].

These results show that skein lasagna modules are not only a boundary-link extension of link homology. They also encode embedded-surface data, produce genus bounds, and in several settings distinguish smooth structures or isotopy classes that are invisible to a purely three-dimensional perspective [2401.06600, 2402.10452].

## 6. Gluing, corners, trisections, and categorical extensions

The gluing structure of skein lasagna theory has been made progressively more explicit. In the cornered theory, if \(Y^3\) is a compact oriented 3-manifold with boundary parametrized by a closed surface \(\Sigma\), and \(P\subset\Sigma\) is a finite signed point set, one defines a category \(\mathcal S(Y,P)\) whose objects are framed tangles in \(Y\) with boundary \(\iota(P)\), and whose morphisms are skein lasagna modules of boundary links in \(Y\times I\). For a 4-manifold with corners \(\partial X=(-Y_1)\cup_\Sigma Y_2\), there is a bimodule-valued functor
\[
F_{X,P}(T_1,T_2)=\mathcal S\bigl(X,\phi(-T_1\cup T_2)\bigr).
\]
The main gluing theorem identifies
\[
F_{X_1,P}\otimes_{\mathcal S(Y_2,P)}F_{X_2,P}\cong F_{X_1\cup_{Y_2}X_2,P},
\]
and a self-gluing theorem relates \(\mathrm{HH}_0(F_{X,P})\) to the skein module of the self-glued 4-manifold with a residual \(P\times S^1\) boundary link. Applied to a trisection \(X=X_1\cup X_2\cup X_3\), this produces a presentation of \(\mathcal S(X)\) in terms of the three standard cornered pieces and a Hochschild-type trace [2512.05861].

A parallel formulation with distinguished 3-manifolds in the boundary gives a tensor-product gluing formula
\[
S^\star(X;T)\cong S^\star(X_1;Y;T_1)\otimes_{S^\star(Y;P)}S^\star(X_2;Y;T_2)
\]
for any functorial link theory \(H^\star\). In that language, 1-handle attachment becomes a Hochschild \(0\)-homology construction, 2-handle attachment becomes tensoring with the \(B^4\) module over the annular algebra of the attaching torus, and 3-handle attachment becomes quotienting by sphere-evaluation relations. The paper notes that a similar construction was introduced independently by Blackwell–Krushkal–Luo [2602.17825].

This cornered viewpoint clarifies a recurring theme of the subject: the skein lasagna module is neither merely a 4-manifold invariant nor merely a decorated link homology. It is a gluing-sensitive object attached to a 4-manifold together with boundary and corner data, and it naturally interacts with Hochschild homology, mapping telescopes, trisection diagrams, and extended TQFT ideas [2206.04616, 2512.05861]. A plausible implication is that future progress will continue to come from translating between these models rather than privileging a single presentation.

Source: https://www.emergentmind.com/topics/skein-lasagna-modules