Internal Skein Modules

• Internal skein modules are refined invariants that retain algebraic structures via internal Hom constructions instead of reducing to scalar outputs.
• They leverage categorical tools like module categories and free cocompletion to incorporate boundary data, enabling excision and tensor-product formulas.
• Applications span deformation quantization, holonomicity of 3-manifolds, and explicit algebraic models linking quantum groups with topological invariants.

Internal skein modules are refinements of ordinary skein modules in which the skein-theoretic output is retained as an internal algebraic object, typically a functor $A^{op}\to \mathrm{Vect}$, an internal endomorphism algebra in a free cocompletion of a skein category, or a boundary-controlled module or bimodule, rather than being immediately reduced to its value at the tensor unit. In three-dimensional skein theory this formalism was developed for ribbon categories by Ben-Zvi–Brochier–Jordan and by Gunningham–Jordan–Safronov; the ordinary skein module is recovered by evaluation at $1$, while the internal object supports excision, tensor-product formulas, and deformation-quantization methods (Gunningham et al., 2019). Later work extended the same philosophy to stated and gated boundary formalisms, derived skein theory, worldsheet and defect skein theories, and four-dimensional skein lasagna modules (Haïoun, 2021, Bai, 9 Jun 2026, Ekholm et al., 2024, Kinnear et al., 5 Jun 2026, Manolescu et al., 2022).

1. Categorical core of the internal construction

For a ribbon category $A$, the internal skein algebra of a punctured surface $\Sigma^*$ is defined as

$$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$

and for a $3$-manifold $N$ with boundary $\partial N\cong \Sigma$, the internal skein module is

$$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$

The ordinary skein module is recovered by evaluation at the tensor unit: $Sk_A(N)\cong Sk_A(N)(1).$ In this formulation the internal skein algebra is an algebra object in the free cocompletion $1$0, and the ordinary skein algebra is its invariant part $1$1 (Gunningham et al., 2019).

A related description, emphasized in the comparison with stated skein theory, defines the internal skein algebra as the internal endomorphism object $1$2 of the empty object in the free cocompletion of the skein category. For a surface with a chosen boundary edge, this internal endomorphism object is characterized by a universal property

$1$3

natural in $1$4. This perspective makes clear that “internal” refers not merely to skeins drawn inside a manifold, but to an internal-Hom construction in a module category over the skein category of the disk or strip (Haïoun, 2021).

A central misconception is that internal skein modules are simply ordinary skein modules with extra marked points. The formalism is stronger: the extra boundary data are not auxiliary decorations but the mechanism by which the skein category becomes a module category and the skein algebra becomes an internal endomorphism algebra. This suggests that internal skein modules are best viewed as coefficient-category-valued enhancements of ordinary skein theory rather than as a minor variant of relative skein modules.

2. Boundary formalisms: stated skeins, multi-edge structures, and gates

The comparison between internal skein algebras and stated skein algebras is explicit in the $1$5 Kauffman bracket setting. For $1$6, the bigon stated skein algebra $1$7 is isomorphic to the coribbon Hopf algebra $1$8, and for a connected surface $1$9 with a chosen boundary edge there is an isomorphism

$A$0

as $A$1-comodule algebras. For marked surfaces with several boundary edges, one obtains multi-edge internal skein algebras $A$2, together with a distinction between left and right boundary edges. The asymmetry is reconciled by a half-twist: the right internal skein algebra is the braided opposite of the left one, and the stated-skein right-edge convention differs from the internal-skein left-edge convention by precisely this half-twist (Haïoun, 2021).

Excision survives in the multi-edge setting. If two marked surfaces are glued along matching right and left boundary collars, the internal skein algebra of the glued surface is the invariant part of the tensor product of the two internal skein algebras. The same paper shows that the free cocompletion of the reduced skein category is equivalent to modules over the multi-edge internal skein algebra. This is the categorical source of the gluing behavior that later becomes fundamental in both holonomicity arguments and four-dimensional analogues (Haïoun, 2021).

A later boundary formalism replaces boundary edges by boundary disks or “gates.” For a surface $A$3 with gates $A$4, the $A$5-internal skein algebra is

$A$6

and internal skein modules for bordisms become bimodules over the boundary internal skein algebras. In that formulation internal skein modules are described as functorial avatars of stated skein modules (Jordan et al., 26 Sep 2025).

3. Excision, Heegaard splittings, finiteness, and holonomicity

The decisive structural result for three-dimensional internal skein modules is the tensor-product formula coming from Heegaard splitting. If

$A$7

then

$A$8

When $A$9 and $\Sigma^*$0 is not a root of unity, the invariant part is unnecessary: $\Sigma^*$1 For a Heegaard splitting by handlebodies, the formula simplifies further to an ordinary algebraic tensor product over the skein algebra of the surface. The simplification rests on two facts: handlebody skein modules are cyclic over the internal skein algebra, and the $\Sigma^*$2-cycles act trivially because they bound disks in the handlebody. In the same framework one has

$\Sigma^*$3

where $\Sigma^*$4 is the braided function algebra; for $\Sigma^*$5 with $\Sigma^*$6, this is a flat deformation quantization of $\Sigma^*$7 with the Fock–Rosly Poisson structure (Gunningham et al., 2019).

These internal formulas lead to the finiteness theorem for closed $\Sigma^*$8-manifolds: $\Sigma^*$9 and more generally, for connected reductive $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$0 and generic $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$1, the $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$2-skein module $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$3 is finite-dimensional over $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$4. The proof passes through holonomic deformation-quantization modules: the internal skein algebra quantizes a symplectic variety, the handlebody modules quantize Lagrangian subvarieties, and the tensor product of the resulting holonomic modules is finite-dimensional after inverting $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$5 (Gunningham et al., 2019).

A later boundary-inclusive refinement proves that for $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$6 or $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$7 at generic $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$8, internal skein modules with gates are not only finitely generated over the boundary internal skein algebra but holonomic. In the paper’s deformation-quantization sense, holonomicity means finite generation together with Lagrangian singular support, or equivalently minimal Gelfand–Kirillov dimension on the relevant symplectic leaf. The proof introduces skein transfer bimodules arising from compression bodies obtained by $$SkAlg_A(\Sigma^*)\colon A^{op}\to Vect,\qquad V\mapsto Hom_{SkCat_A(\Sigma^*)}(P(V),1),$$9-handle attachments, together with $3$0-analogues of inverse and direct image functors and a Kashiwara-type statement. The main conclusion is that tensoring with these transfer bimodules preserves holonomicity (Jordan et al., 26 Sep 2025).

4. Derived theory and explicit algebraic models

Derived skein theory internalizes the ordinary theory before taking $3$1-th homology. For a ribbon tensor category $3$2, the internal derived skein module of a compact oriented $3$3-manifold $3$4 with boundary is

$3$5

and the internal derived skein algebra of a punctured surface is

$3$6

The axioms require

$3$7

and gluing is governed by a bar construction. In the generic quantum-group case, this leads to a computable Heegaard formula

$3$8

while for $3$9 one gets the Hochschild formula

$N$0

The same paper identifies

$N$1

and proves finite-dimensionality of the homologies of closed derived skein modules for generic $N$2 (Bai, 9 Jun 2026).

Explicit internal endomorphism algebras also appear in higher-rank skein theory on the torus. For $N$3, the relative skein algebras of $N$4 are

$N$5

and for $N$6, the determinant correction forces $N$7. The skein category of $N$8 is Morita equivalent to finitely presented algebras generated by these distinguished objects. In the Schur–Weyl case $N$9, the paper constructs an isomorphism between the $\partial N\cong \Sigma$0-point relative skein algebra and a specialized double affine Hecke algebra, with the consequence that every tangle class in the relative $\partial N\cong \Sigma$1-point skein algebra is a linear combination of braids modulo skein relations. The same framework yields

$\partial N\cong \Sigma$2

and

$\partial N\cong \Sigma$3

obtained through Hochschild homology of the torus skein category (Gunningham et al., 2024).

These results show that internal skein modules are not merely abstract categorical enhancements. They also furnish explicit algebraic models—quantum differential operators, quantum coordinate algebras, DAHA specializations, and Hochschild constructions—that convert topological skein questions into tractable algebra.

5. Geometric reinterpretations: worldsheets and defects

A distinct but related use of the internal skein philosophy appears in symplectic and enumerative geometry. The worldsheet skein module replaces link generators by holomorphic worldsheets: $\partial N\cong \Sigma$4 where $\partial N\cong \Sigma$5 is a properly embedded oriented $\partial N\cong \Sigma$6-submanifold that is locally Lagrangian. The module is generated by isotopy classes of such surfaces modulo two local skein relations, the hyperbolic and elliptic relations, which are worldsheet analogues of the HOMFLYPT skein relations. In this setting the worldsheet skein module is described as the universal target for skein-valued open curve counts, and the corresponding worldsheet skein $\partial N\cong \Sigma$7-module is the quotient by recursion operators. For the negative Hopf link conormal, the worldsheet skein $\partial N\cong \Sigma$8-module is generated by three operator polynomials, and for any Lagrangian filling the corresponding partition function is the unique solution of the resulting annihilation equations with prescribed initial condition. The same paper proves that the skein-valued partition functions admit quiver-like expansions generated by a small collection of basic disks and annuli and their multiple covers (Ekholm et al., 2024).

Another extension introduces line and point defects on the boundary of a $\partial N\cong \Sigma$9-manifold. Here the boundary $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$0-strata are labeled by right $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$1-module categories and the point defects by $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$2-module functors between relative tensor products of those module categories. The defect skein module $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$3 is the quotient of the free vector space on defect-colored ribbon graphs by local relations read off from valid cubes adapted to the defect stratification. In the semisimple case the construction recovers the defect Reshetikhin–Turaev state space: $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$4 The local relations therefore globalize not only the graphical calculus of a ribbon category but also the graphical calculus of module categories and module functors (Kinnear et al., 5 Jun 2026).

These geometric versions make the term “internal” genuinely plural. In one lineage it refers to internal endomorphism algebras in free cocompletions; in another it denotes a skein theory built from surfaces, holomorphic curves, or defect-labeled ribbon graphs internal to a symplectic or TQFT background. The convergence lies in universality: each construction produces a target in which local skein relations encode the geometry of a richer moduli problem.

6. Four-dimensional internalizations and applications

In four dimensions the relevant objects are skein lasagna modules. For a smooth, compact, oriented $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$5-manifold $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$6 and a framed oriented link $$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$7, the skein lasagna module

$$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$8

is generated by lasagna fillings

$$Sk_A(N)\colon A^{op}\to Vect,\qquad V\mapsto SkMod_A(N,P(V)).$$9

where $Sk_A(N)\cong Sk_A(N)(1).$0 is a properly embedded oriented framed surface in $Sk_A(N)\cong Sk_A(N)(1).$1, the $Sk_A(N)\cong Sk_A(N)(1).$2 are embedded $Sk_A(N)\cong Sk_A(N)(1).$3-balls, and the labels $Sk_A(N)\cong Sk_A(N)(1).$4 lie in Khovanov–Rozansky homology. This invariant extends $Sk_A(N)\cong Sk_A(N)(1).$5 from $Sk_A(N)\cong Sk_A(N)(1).$6, since

$Sk_A(N)\cong Sk_A(N)(1).$7

A handle-decomposition analysis shows that $Sk_A(N)\cong Sk_A(N)(1).$8-handles lead to Hochschild homology of $Sk_A(N)\cong Sk_A(N)(1).$9-ball tangle categories, $1$00-handles to cabling together with braid invariance and dot relations, $1$01-handles to coequalizer relations, and $1$02-handles act trivially. In the example

$1$03

the module is infinite-dimensional for $1$04 when $1$05, showing that the four-dimensional theory can fail local finite-dimensionality even when three-dimensional skein modules satisfy strong finiteness theorems (Manolescu et al., 2022).

A more flexible $1$06-dimensional internalization assigns modules not only to pairs $1$07 but to triples $1$08, where $1$09 is a distinguished boundary $1$10-manifold and $1$11 a boundary tangle. The associated triple module

$1$12

is a module over an algebra $1$13, and gluing along $1$14 is expressed by the bimodule tensor product

$1$15

In this formulation $1$16-handle attachment becomes Hochschild-type gluing, $1$17-handle attachment becomes a cabling-and-quotient construction, and $1$18-handle attachment becomes quotienting by the attaching-region algebra action. This packages and generalizes formulas due to Chen, Manolescu–Neithalath, Manolescu–Walker–Wedrich, and Ren–Willis (Martin et al., 19 Feb 2026).

A Bar–Natan variant of skein lasagna modules replaces Khovanov–Rozansky input by Bar–Natan homology over $1$19. The resulting module $1$20 carries an $1$21-action in which multiplication by $1$22 corresponds to attaching an unknotted $1$23-handle to a connected component of the surface. This makes internal stabilization algebraic and allows the construction of exotic surface pairs whose difference has $1$24-torsion order $1$25, so that one internal stabilization is not enough to unknot them (Sullivan, 4 Apr 2025).

The four-dimensional theory therefore extends the internal skein idea from links and tangles in $1$26-manifolds to surfaces in $1$27-manifolds, together with categorical boundary labels and gluing algebras. A plausible implication is that “internal skein module” has become a unifying label for several constructions in which skein relations are reorganized as module-theoretic or categorical structures attached to codimension-one or codimension-two boundary data, rather than as scalar reductions of link isotopy alone.