Internal Skein Algebra

• Internal skein algebra is a framework that internalizes skein theory by defining algebra objects as endomorphisms within categorical and homological structures.
• It organizes constructions from factorisation homology, quantum group representations, and boundary-relative states into a unified local-to-global formalism.
• The theory further refines skein algebras with deformation quantization, derived and holonomic enhancements, and cluster and annular model variations.

Searching arXiv for papers on internal skein algebra, skein categories, stated skein, and related factorization-homology frameworks. “Internal skein algebra” is not a single universally standardized term in the arXiv literature. Instead, it denotes a family of closely related constructions in which skein theory is organized internally to a larger algebraic or categorical environment: as an endomorphism algebra inside a skein category, as an algebra object in a braided or ribbon tensor category obtained from factorisation homology, as a boundary-relative or stated skein algebra with endpoint data, or as the algebra over which internal skein modules of 3-manifolds are defined. In this sense, the topic is best understood as a convergence point of skein categories, quantum group representation theory, factorisation homology, stated skein theory, and deformation quantization (Cooke, 2019, Bai, 9 Jun 2026, Jordan et al., 26 Sep 2025).

1. Terminology and scope

A precise starting point is that several foundational papers explicitly do not use the exact phrase “internal skein algebra,” while nevertheless providing the closest rigorous notions. In Cooke’s framework, the ordinary skein algebra of a surface is recovered as the endomorphism algebra of the empty object in a skein category, and after free cocompletion one obtains an algebra object $A_\Sigma$ internal to the coefficient tensor category (Cooke, 2019). In the derived theory, the internal skein algebra of a punctured surface is the monad attached to the empty labeling and reconstructs the derived skein category by monadicity (Bai, 9 Jun 2026). In the boundary-relative setting with gates, the internal skein algebra is the internal endomorphism algebra of the empty object in a skein category regarded as a module category over the tensor category attached to the gates (Jordan et al., 26 Sep 2025).

The literature therefore supports three principal meanings. The first is categorical: the ordinary skein algebra is an endomorphism algebra internal to a skein category. The second is factorisation-homological: collars and punctured surfaces determine algebra objects and module categories inside $Cat_k^\times$, $LFP_k$, or derived enhancements thereof. The third is boundary-relative: skeins are allowed to meet the boundary through marked intervals, states, or gates, and the resulting internal or stated skein algebra governs gluing, modules, and representation theory. A persistent misconception is that these are competing definitions; the cited papers instead present them as different levels of the same local-to-global formalism (Cooke, 2019, Matson, 1 Mar 2025).

2. Skein categories, collars, and factorisation homology

For a strict $k$-linear ribbon category $\mathscr V$ and an oriented smooth finitary surface $\Sigma$, the skein category $Sk_{\mathscr V}(\Sigma)$ is obtained from the category of $\mathscr V$-coloured ribbon diagrams in $\Sigma\times[0,1]$ by quotienting by local evaluation relations coming from the ribbon functor

$eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$

Its objects are finite sets of coloured framed points in $Cat_k^\times$0, and its morphisms are $Cat_k^\times$1-linear combinations of coloured ribbon tangles modulo isotopy and local skein relations. The base case

$Cat_k^\times$2

identifies the skein category of the disk with the coefficient category itself (Cooke, 2019).

Within this category, the ordinary skein algebra is recovered as

$Cat_k^\times$3

This is the first rigorous sense in which the skein algebra is “internal”: it is not introduced as an external algebra of closed links, but as the endomorphism algebra of a distinguished object. When $Cat_k^\times$4 is a collar, $Cat_k^\times$5 is monoidal, with unit $Cat_k^\times$6, and surfaces with boundary become module categories over this collar skein category. If $Cat_k^\times$7 and $Cat_k^\times$8 are glued along $Cat_k^\times$9, excision takes the form

$LFP_k$0

with the tensor product understood as Tambara’s relative tensor product of $LFP_k$1-linear categories (Cooke, 2019).

Cooke’s main theorem identifies the skein category itself with $LFP_k$2-linear factorisation homology: $LFP_k$3 The cylinder then plays the role of an $LFP_k$4-algebra object, and surface skein categories are corresponding module objects. This places “internal skein algebra” squarely inside the ambient local-to-global formalism of factorisation homology, rather than treating it as ad hoc terminology (Cooke, 2019).

3. Internal endomorphism algebras of punctured surfaces

After passage from small skein categories to cocomplete settings, the internal viewpoint becomes stronger. For punctured surfaces and coefficients in a rigid abelian balanced braided category $LFP_k$5, factorisation homology produces an $LFP_k$6-module category equivalent to modules over an algebra object

$LFP_k$7

Here $LFP_k$8 is the distinguished object defined by the empty embedding, and one has

$LFP_k$9

This is the strongest literal sense of internal skein algebra in Cooke’s framework: not merely an endomorphism ring in a category, but an algebra object internal to the coefficient tensor category (Cooke, 2019).

For $k$0, with $k$1 connected and semisimple and $k$2 equal to $k$3 or not a root of unity, Cooke proves for punctured surfaces that the ordinary skein algebra coincides with the invariant algebra of this internal object: $k$4 Combined with the theorem of Ben-Zvi–Brochier–Jordan that the invariant algebra is a deformation quantisation of the character variety $k$5, this identifies the punctured-surface skein algebra with a deformation quantisation of $k$6 (Cooke, 2019).

The derived theory refines this picture. For a ribbon tensor category $k$7 and a punctured surface $k$8, the internal derived skein algebra is defined by

$k$9

and satisfies

$\mathscr V$0

In the semisimple case, especially $\mathscr V$1 with $\mathscr V$2 not a root of unity, the derived internal skein algebra collapses to the ordinary internal skein algebra: $\mathscr V$3 Moreover, for $\mathscr V$4,

$\mathscr V$5

with braided tensor product understood in the coefficient category (Bai, 9 Jun 2026).

4. Boundary, states, and gates

A second major branch of the subject internalizes skein theory by allowing controlled boundary interaction. In stated skein theory, one works with tangles in $\mathscr V$6 whose endpoints lie on marked boundary arcs and carry states in $\mathscr V$7. The stated skein algebra $\mathscr V$8 is defined by the Kauffman bracket relations together with trivial arc, state exchange, and height exchange relations; it extends the ordinary Kauffman bracket skein algebra by an embedding

$\mathscr V$9

The thesis literature emphasizes that the main reason stated skein modules were created was the splitting theorem: boundary states make cutting and gluing compatible with algebraic reconstruction (Matson, 1 Mar 2025).

This boundary-relative formalism is identified explicitly with internal skein algebra in the factorisation-homological sense. For a once-punctured surface $\Sigma$0 with one marking, the stated skein algebra is identified in the literature with the internal skein algebra $\Sigma$1 as an $\Sigma$2-comodule algebra (Matson, 1 Mar 2025). A closely related reduction appears in the skein-theoretic construction of quantum duality maps, where the reduced stated skein algebra

$\Sigma$3

and its congruent subalgebra serve as the surface-attached algebraic objects through which quantum traces and duality maps factor (Ishibashi et al., 2023).

The gated version makes the internality completely explicit. Let $\Sigma$4 be a surface with boundary and let

$\Sigma$5

be a finite collection of disjoint gates in annular neighborhoods of the boundary. Writing $\Sigma$6, the $\Sigma$7-internal skein algebra is defined by

$\Sigma$8

If $\Sigma$9, one recovers the ordinary skein algebra; if $Sk_{\mathscr V}(\Sigma)$0 consists of a single gate, one recovers the internal skein algebra of Gunningham–Jordan–Safronov type. A 3-manifold with gated boundary then determines an internal skein bimodule over the internal skein algebras of its incoming and outgoing boundary surfaces, and gluing is expressed by relative tensor product over the intermediate internal skein algebra (Jordan et al., 26 Sep 2025).

This boundary-relative theory supports a strong finiteness theorem. For generic $Sk_{\mathscr V}(\Sigma)$1 and $Sk_{\mathscr V}(\Sigma)$2, $Sk_{\mathscr V}(\Sigma)$3, or $Sk_{\mathscr V}(\Sigma)$4, internal skein modules of 3-manifolds with boundary are finitely generated and holonomic over the internal skein algebra of the boundary. In the standard one-gate-per-boundary-component case, holonomicity is expressed by the Gelfand–Kirillov dimension formula

$Sk_{\mathscr V}(\Sigma)$5

equivalently by a Lagrangian singular support condition on the open symplectic leaf of the Fock–Rosly Poisson variety (Jordan et al., 26 Sep 2025).

5. Root-of-unity structure, centers, and central reductions

A different but complementary meaning of “internal” arises in root-of-unity Kauffman bracket theory, where the emphasis shifts from categorical internal Homs to the internal structure of the algebra itself: its center, filtrations, local bases, and central localizations. For a punctured oriented surface $Sk_{\mathscr V}(\Sigma)$6, specialized at a primitive $Sk_{\mathscr V}(\Sigma)$7-th root of unity with $Sk_{\mathscr V}(\Sigma)$8 odd, the Kauffman bracket skein algebra $Sk_{\mathscr V}(\Sigma)$9 contains a central character subalgebra

$\mathscr V$0

canonically identified with the coordinate ring of the $\mathscr V$1-character variety. For punctured surfaces,

$\mathscr V$2

The algebra is finite over $\mathscr V$3, residues modulo $\mathscr V$4 control the noncentral part, and after localizing at nonzero central character elements the algebra becomes a division algebra. After suitable localization over the full center, it splits as

$\mathscr V$5

where $\mathscr V$6 and $\mathscr V$7 are commutative subalgebras generated by two pants decompositions (Frohman et al., 2016).

A related root-of-unity result proves that for noncompact finite-type surfaces the central localization

$\mathscr V$8

is a symmetric Frobenius algebra over the function field $\mathscr V$9. The normalized trace of left multiplication gives the Frobenius form, and primitive diagrams generate finite field extensions inside the localized algebra (Abdiel et al., 2015). In this line of work, “internal skein algebra” means that the skein algebra is studied through its own internal central and trace-theoretic organization rather than through external representations.

Further refinements impose central character conditions directly. The sliced skein algebra

$\Sigma\times[0,1]$0

fixes the scalar values of the peripheral curves. It remains a domain over a commutative domain ground ring, and at roots of unity its center is described by the Chebyshev–Frobenius image

$\Sigma\times[0,1]$1

with PI-degree

$\Sigma\times[0,1]$2

Smooth points of the sliced character variety are Azumaya for the sliced skein algebra, and their preimages are Azumaya for the full skein algebra (Frohman et al., 2023).

At the level of a unified formalism, LRY skein algebras encompass Kauffman bracket, Roger–Yang, and stated skein algebras. They are domains over arbitrary commutative domain ground rings, are orderly finitely generated, are Noetherian when the ground ring is, and admit filtrations whose associated graded algebras are monomial subalgebras of quantum tori. In the pants-decomposition setting, the leading product formula takes the form

$\Sigma\times[0,1]$3

making the internal multiplication law visible as a highest-term quantum-torus rule (Bloomquist et al., 2023).

6. Arc, annular, and cluster-enhanced models

Internalization can also occur by enlarging the class of generators from closed links to arcs and relative tangles. The arc-and-link skein algebra $\Sigma\times[0,1]$4 of a punctured surface includes framed arcs ending at punctures and satisfies, besides the Kauffman bracket skein relation and the trivial loop relation, a puncture-skein relation and a puncture-loop relation. It is a topologically free associative $\Sigma\times[0,1]$5-algebra, its classical limit is the Poisson algebra $\Sigma\times[0,1]$6 of curves and arcs, and the quantization map to decorated Teichmüller space is a Poisson algebra homomorphism (Roger et al., 2011). Roger–Yang’s later punctured-surface skein algebra $\Sigma\times[0,1]$7 sharpens this picture: $\Sigma\times[0,1]$8, and for locally planar triangulations the map from the curve algebra to functions on decorated Teichmüller space is injective, implying that both the classical curve algebra $\Sigma\times[0,1]$9 and the quantum algebra $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$0 are domains (Moon et al., 2019).

The annular case exhibits an explicitly categorical internalization. The skein category $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$1 of the annulus is equivalent to the affine Temperley–Lieb category, and the extended affine Temperley–Lieb algebra is identified with an internal endomorphism algebra

$eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$2

An arc-insertion endofunctor of $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$3 induces algebra maps $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$4, and in the trivial Dehn twist specialization the direct sum of link-pattern spaces becomes a graded associative algebra that may be regarded as a relative version of the Roger–Yang skein algebra of arcs and links on the punctured disc (Qasimi et al., 2017).

A further internalization occurs in skein-theoretic cluster theory for HOMFLYPT skeins. There the completed surface skein algebra $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$5 carries a mutation formalism in which the ordinary quantum dilogarithm is replaced by a skein Baxter operator

$eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$6

and mutation is defined by conjugation by $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$7 together with a topological lattice mutation. On the closed torus one has the pentagon identity

$eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$8

which is the skein-theoretic analogue of the quantum cluster pentagon relation (Hu et al., 2023).

These constructions connect directly to reduced stated skein algebras. Quantum duality maps for general marked surfaces factor through the reduced stated skein algebra and its congruent subalgebra, using quantum trace maps and skein lifts of integral $eval: Ribbon_{\mathscr V}([0,1]^2)\to \mathscr V.$9-laminations. In this setting the reduced stated skein algebra is the internal surface algebra mediating between topological laminations and triangulation-dependent quantum cluster tori (Ishibashi et al., 2023).

7. Derived and holonomic directions

The derived theory extends internal skein algebra from ordinary algebra objects to DG and homological settings. The derived skein module $Cat_k^\times$00 is characterized by the skein property

$Cat_k^\times$01

and by an excision axiom expressed through a simplicial bar construction. For a punctured surface $Cat_k^\times$02, the internal derived skein algebra $Cat_k^\times$03 controls gluing by derived tensor product; for a Heegaard splitting $Cat_k^\times$04,

$Cat_k^\times$05

The Hochschild formula

$Cat_k^\times$06

identifies $Cat_k^\times$07 with the Hochschild homology of the derived skein category (Bai, 9 Jun 2026).

An important clarification follows from semisimplicity. For semisimple ribbon categories, the internal algebraic input itself does not acquire extra derived corrections: $Cat_k^\times$08 The new higher information appears only after taking derived relative tensor products and then filling the 3-handle. Thus derived skein theory is not obtained by naively deriving the underived skein algebra of the same manifold; the internal punctured-surface algebra remains ordinary in the semisimple generic case, while the closed-manifold invariant can detect higher homological data such as $Cat_k^\times$09 and $Cat_k^\times$10 (Bai, 9 Jun 2026).

A parallel development concerns holonomicity. Internal skein algebras of gated boundary surfaces admit explicit $Cat_k^\times$11-difference-operator presentations, and transfer bimodules associated to 2-handle attachments act as $Cat_k^\times$12-analogues of D-module transfer bimodules. Using Ore localization, Koszul resolutions, and a Kashiwara-type statement, one proves that transfer preserves holonomicity and hence that internal skein modules of 3-manifolds with boundary are holonomic over the boundary internal skein algebra (Jordan et al., 26 Sep 2025).

Taken together, these developments suggest a stable conceptual picture. The internal skein algebra is the local algebraic object assigned to a punctured, bordered, or gated surface; ordinary skein algebras, stated skein algebras, annular endomorphism algebras, and internal moduli algebras are its various incarnations; and derived, cluster, and holonomic theories refine rather than replace this local core (Cooke, 2019, Jordan et al., 26 Sep 2025).