Skein Lasagna Modules: 4D Link Homology

• Skein lasagna modules are a 4-dimensional extension of link homology that combine framed surfaces with labeled input balls via local replacement relations.
• They use colimit constructions and cabled handlebody methods to yield explicit computations, tensor-product formulas, and refined TQFT state spaces.
• Their framework facilitates genus bounds and detects exotic smooth structures, advancing practical tools in 4-manifold topology and link theory.

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 (Manolescu et al., 2020, Manolescu et al., 2022, Morrison et al., 2024, Kauffman et al., 13 Feb 2026).

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 $KhR_N$0, and the elementary presentation is designed so that the degree-zero blob differential is built into the relations (Manolescu et al., 2020).

The grading conventions depend on the chosen input theory. In one $KhR_N$1 normalization, a filling $KhR_N$2 carries

$KhR_N$3

while other formulations record additionally the relative homology class $KhR_N$4, or treat the skein module as the blob-degree-zero piece of a triply graded theory (Manolescu et al., 2020, Morrison et al., 2024). 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 $KhR_N$5 rel $KhR_N$6, the morphisms are embedded cobordisms between skeins, and a functor $KhR_N$7 sends each input link to its link-homology module. The skein lasagna module is then

$KhR_N$8

so concretely $KhR_N$9 whenever a morphism $(W;L)$0 is present (Kauffman et al., 13 Feb 2026). In the earlier handle-decomposition work, this same structure is interpreted as the “Hilbert space” of a fully-extended $(W;L)$1-dimensional TQFT, with ball-replacement functioning as the local relation that makes gluing well defined (Manolescu et al., 2022).

2. Handle decompositions, cabled formulas, and homotopy colimits

For 4-manifolds built from $(W;L)$2 by 2-handle attachment, Manolescu–Neithalath give a concrete cabled description. If $(W;L)$3 is obtained by attaching $(W;L)$4 2-handles along a framed link $(W;L)$5, and $(W;L)$6, they define a cabled Khovanov–Rozansky complex $(W;L)$7 by summing over links $(W;L)$8, then quotienting by the braid-group actions on the parallel strands and by dot-annihilation relations. The resulting theorem is a canonical isomorphism

$(W;L)$9

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 $B_i\subset \operatorname{Int} W$0 (Manolescu et al., 2020).

The broader handle-decomposition formalism of Manolescu–Walker–Wedrich starts with a Morse filtration $B_i\subset \operatorname{Int} W$1, where $B_i\subset \operatorname{Int} W$2 is the 0- and 1-handlebody, $B_i\subset \operatorname{Int} W$3 adds 2-handles, and $B_i\subset \operatorname{Int} W$4 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 $B_i\subset \operatorname{Int} W$5; 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 $B_i\subset \operatorname{Int} W$6 into an explicit sequence of cabling, quotienting, and coequalizing operations tied directly to a Kirby diagram (Manolescu et al., 2022).

A later reformulation replaces the ordinary colimit by a homotopy colimit. In the setting of $B_i\subset \operatorname{Int} W$7, 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 $B_i\subset \operatorname{Int} W$8, and its totalization $B_i\subset \operatorname{Int} W$9 is identified with the homotopy colimit $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$0. Proposition 2.12 in that paper shows that this totalization satisfies the universal property of the colimit in the completed Karoubian/Bar-Natan category (Sullivan et al., 2024). 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 $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$1 theories. A uniform theory for any functorial link invariant $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$2 in $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$3 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 $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$4 with the evaluation map on the trivial sphere (Martin et al., 19 Feb 2026). 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 $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$5, represented by a single 0-framed unknot. For general $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$6,

$$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$7

with $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$8 for $$\Sigma\subset W\setminus \bigcup_i \operatorname{Int} B_i$$9, and the summand in relative class $\partial \Sigma=L$0 is the subspace of homogeneous polynomials of total degree $\partial \Sigma=L$1. In the $\partial \Sigma=L$2 case this yields a single copy of $\partial \Sigma=L$3 in bidegrees $\partial \Sigma=L$4 with $\partial \Sigma=L$5 and vanishing otherwise (Manolescu et al., 2020). This computation became the base case for several later refinements.

A more specialized computation concerns $\partial \Sigma=L$6, where $\partial \Sigma=L$7 is a geometrically essential boundary link. The key tool is the categorified Jones–Wenzl projector $\partial \Sigma=L$8, defined as the filtered colimit $\partial \Sigma=L$9. It is idempotent up to homotopy, kills turnbacks, and “eats” braid generators. In the cabling telescope one inserts $v_i\in KhR_N(B_i,L_i)$0 on the core strands before capping off by the belt; Proposition 5.4 then shows that only the through-degree-0 Rozansky projector $v_i\in KhR_N(B_i,L_i)$1 survives when $v_i\in KhR_N(B_i,L_i)$2 is even, while the whole telescope is annihilated when $v_i\in KhR_N(B_i,L_i)$3 is odd. The resulting theorem is that

• if $v_i\in KhR_N(B_i,L_i)$4 is odd or $v_i\in KhR_N(B_i,L_i)$5 is odd, then $v_i\in KhR_N(B_i,L_i)$6;
• if $v_i\in KhR_N(B_i,L_i)$7 and $v_i\in KhR_N(B_i,L_i)$8 is even, then

$v_i\in KhR_N(B_i,L_i)$9

The same paper identifies $L_i=\Sigma\cap \partial B_i$0 with $L_i=\Sigma\cap \partial B_i$1, and relates the projector term to the Rozansky–Willis invariant for nullhomologous links in $L_i=\Sigma\cap \partial B_i$2 (Sullivan et al., 2024).

The same projector analysis yields a vanishing theorem for $L_i=\Sigma\cap \partial B_i$3. Using the 0-framed Hopf link Kirby diagram, the module in class $L_i=\Sigma\cap \partial B_i$4 is viewed as an iterated colimit in the two cable directions. If either $L_i=\Sigma\cap \partial B_i$5 or $L_i=\Sigma\cap \partial B_i$6 is odd, the intermediate colimit already vanishes. When both are even, the dotted annulus map on $L_i=\Sigma\cap \partial B_i$7 is analyzed, and the relevant obstruction classes in $L_i=\Sigma\cap \partial B_i$8 vanish or become nilpotent. The conclusion is

$L_i=\Sigma\cap \partial B_i$9

for all levels, confirming a conjecture of Manolescu (Sullivan et al., 2024).

The theory is not uniformly finite-dimensional. For $v_i$0 over a perfect field $v_i$1, Manolescu–Walker–Wedrich compute

$v_i$2

and in fact the bidegree $v_i$3 summand is already infinite-dimensional when $v_i$4 (Manolescu et al., 2022). A common misconception is therefore that skein lasagna modules should behave like finite-rank TQFT state spaces in every local example; the explicit $v_i$5 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 $v_i$6	$v_i$7	non-vanishing, genus bound, deformation decomposition
Bar-Natan	$v_i$8 over $v_i$9	dot-trading, neck-cutting, $v_i$0-torsion gluing behavior
Floer lasagna	$v_i$1	2-handlebody model via cabled link Floer homology
1-dimensional inputs	Rozansky–Willis/Khovanov in $v_i$2	handlebody model and lasso relation
Stable-homotopy refinement	Lipshitz–Sarkar spectra $v_i$3	cohomology recovers $v_i$4, but spectrum is stronger for $v_i$5

For equivariant and deformed $v_i$6 theories, Morrison–Walker–Wedrich define $v_i$7 using $v_i$8, $v_i$9, or $KhR_N$00, with decomposition over $KhR_N$01. In the $KhR_N$02-equivariant theory, if $KhR_N$03 is smoothly embedded, oriented, and homologically diverse, then its lasagna class $KhR_N$04 is non-torsion over $KhR_N$05. They also define

$KhR_N$06

and derive the bound

$KhR_N$07

In the deformed theory, a full decomposition theorem splits $KhR_N$08 over colorings of the boundary-link components by the deformation parameters (Morrison et al., 2024).

The Bar-Natan version replaces $KhR_N$09 by the Frobenius pair $KhR_N$10, $KhR_N$11. Its local relations are birth/death of a small sphere, the dotted sphere relation, dot-trading, and neck-cutting. The resulting module $KhR_N$12 is an $KhR_N$13-module, and connect-sum gluing maps preserve $KhR_N$14-torsion order on a half-torsion-free submodule. The paper computes

$KhR_N$15

and

$KhR_N$16

(Sullivan, 4 Apr 2025).

Floer lasagna modules replace Khovanov–Rozansky homology by $KhR_N$17 and decorate the filling surface with dividing arcs cutting it into $KhR_N$18 and $KhR_N$19. The resulting module $KhR_N$20 is graded by the relative class in $KhR_N$21, Maslov degree $KhR_N$22, and Alexander degree $KhR_N$23. For 4-manifolds obtained by attaching 2-handles to $KhR_N$24, Chen proves a natural bigrading-preserving isomorphism between $KhR_N$25 and a cabled link Floer homology $KhR_N$26 (Chen, 2022).

The 1-dimensional-input Khovanov theory allows input 1-handlebodies rather than only 4-balls, and uses Rozansky–Willis homology in connected sums of $KhR_N$27. If $KhR_N$28, then

$KhR_N$29

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 $KhR_N$30. This framework yields explicit disk-bundle computations, including

$KhR_N$31

as well as vanishing for $KhR_N$32 when $KhR_N$33 and partial vanishing for $KhR_N$34 in positive homological degree (Ren et al., 6 Oct 2025, Montague et al., 28 Jun 2026).

A stable-homotopy refinement replaces the module-valued functor with the Lipshitz–Sarkar spectrum $KhR_N$35 and defines

$KhR_N$36

Its reduced cohomology recovers the Khovanov skein lasagna module,

$KhR_N$37

but for $KhR_N$38 the spectrum is stronger than the $KhR_N$39 skein lasagna module because it retains higher operations such as Steenrod squares (Kauffman et al., 13 Feb 2026).

A further enrichment appears in the equivariant theory over $KhR_N$40, where the module $KhR_N$41 carries a well-defined action of $KhR_N$42. The construction uses green-dotted boundary data satisfying Euler-characteristic constraints, together with the infinitesimal $KhR_N$43-action on equivariant $KhR_N$44-homology and foam complexes (Qi et al., 3 Apr 2026).

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

A central point of skein lasagna theory is that a properly embedded oriented surface $KhR_N$45 with $KhR_N$46 determines a canonical lasagna class. In the equivariant/deformed $KhR_N$47 framework, this class depends only on the relative homology class $KhR_N$48, and has tridegree

$KhR_N$49

The non-vanishing theorem for homologically diverse surfaces then makes the skein module a source of genus bounds. In particular, the quantity $KhR_N$50 functions as a lasagna analogue of Rasmussen’s invariant and reproduces the classical Rasmussen bound when $KhR_N$51, $KhR_N$52, and $KhR_N$53 is a knot (Morrison et al., 2024).

The $KhR_N$54 theory admits a Lee deformation and a lasagna $KhR_N$55-invariant. In that setting,

$KhR_N$56

and for $KhR_N$57,

$KhR_N$58

This invariant satisfies normalization, symmetries, connected-sum and gluing properties, and an adjunction-type genus bound

$KhR_N$59

The same paper applies the $KhR_N$60 skein lasagna module to the exotic pair of knot traces $KhR_N$61 and $KhR_N$62. In relative class $KhR_N$63, one finds

$KhR_N$64

Hence $KhR_N$65, giving what the paper describes as the first analysis-free proof of the existence of exotic compact orientable 4-manifolds (Ren et al., 2024).

The Bar-Natan theory detects exotic surfaces with boundary rather than only closed-manifold phenomena. Hayden’s exotic pair $KhR_N$66 determines a nonzero $KhR_N$67-torsion difference class $KhR_N$68 of order at least $KhR_N$69 in $KhR_N$70. If $KhR_N$71 is primitive, Sullivan proves that after taking a connect sum with $KhR_N$72, the resulting difference class in $KhR_N$73 has the same $KhR_N$74-torsion order as $KhR_N$75. The conclusion is that one internal stabilization is generally not enough for these exotic knotted surfaces (Sullivan, 4 Apr 2025).

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 (Morrison et al., 2024, Ren et al., 2024).

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 $KhR_N$76 is a compact oriented 3-manifold with boundary parametrized by a closed surface $KhR_N$77, and $KhR_N$78 is a finite signed point set, one defines a category $KhR_N$79 whose objects are framed tangles in $KhR_N$80 with boundary $KhR_N$81, and whose morphisms are skein lasagna modules of boundary links in $KhR_N$82. For a 4-manifold with corners $KhR_N$83, there is a bimodule-valued functor

$KhR_N$84

The main gluing theorem identifies

$KhR_N$85

and a self-gluing theorem relates $KhR_N$86 to the skein module of the self-glued 4-manifold with a residual $KhR_N$87 boundary link. Applied to a trisection $KhR_N$88, this produces a presentation of $KhR_N$89 in terms of the three standard cornered pieces and a Hochschild-type trace (Blackwell et al., 5 Dec 2025).

A parallel formulation with distinguished 3-manifolds in the boundary gives a tensor-product gluing formula

$KhR_N$90

for any functorial link theory $KhR_N$91. In that language, 1-handle attachment becomes a Hochschild $KhR_N$92-homology construction, 2-handle attachment becomes tensoring with the $KhR_N$93 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 (Martin et al., 19 Feb 2026).

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 (Manolescu et al., 2022, Blackwell et al., 5 Dec 2025). A plausible implication is that future progress will continue to come from translating between these models rather than privileging a single presentation.