Compatibility of quantum trace and UV-IR maps

Samuel Panitch; Sunghyuk Park

REVIEW 3 major objections 4 minor 38 references

Compatibility of quantum trace and UV-IR maps

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Quantum trace map is the UV-IR map plus one evaluation step

desk verdict A serious, mostly credible paper that proves the Neitzke–Yan conjecture and builds the 3d compatibility square, but Theorem C is overbroad as stated because its homology hypothesis does not guarantee the angle structure needed for the UV-IR map. read the letter →

arxiv 2509.09100 v1 pith:CGQ2I7CQ submitted 2025-09-11 math.GT math.QA

Samuel Panitch , Sunghyuk Park This is my paper · ORCID

classification math.GTmath.QA MSC 57K1657K31

keywords skeinmodulesquantumtracemapUV-IRbrancheddoublecoverTeichmullerspacegluingmodulegeneralizedanglestructuresPachnermoves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that two independent abelianization maps for skein modules—the algebraic quantum trace map and the geometric quantum UV-IR map built from foliations—are the same up to a fixed evaluation step. The main theorem produces a commutative square for ideally triangulated 3-manifolds equipped with a generalized angle structure: after passing from gl2-skeins to sl2-tensor-gl1-skeins, the UV-IR map followed by evaluation equals the quantum trace map tensored with the identity. For surfaces, the same square settles the previously conjectured relation between the 2d versions of these maps. A corollary is that, when a mild intersection-pairing condition holds—for example on knot complements—the 3d quantum trace map can be completely recovered from the UV-IR map, giving a new construction of the 3d quantum trace invariant.

What carries the argument

The load-bearing machinery is a commutative square whose four corners are skein modules: top-left gl2-skein of Y, top-right gl1-skein of the branched double cover, bottom-left sl2-skein of Y tensor gl1-skein of Y, bottom-right square-root quantum gluing module tensor gl1-skein. The top arrow is the 3d quantum UV-IR map, built from a WKB foliation whose leaf space carries the link diagram; singular leaves form the spectral network, and the map sends a framed oriented link to a weighted sum of lifts via direct lifts, detours, and exchanges. The left arrow is the gl2-to-sl2 map π, which factors each gl2 tangle into an sl2 tangle and a gl1 tangle with a boundary sign (-1)^{b(L)}. The right arrow

What would settle it

Compute both sides of π ∘ F_T([L]) = (Tr_T ⊗ id) ∘ ev([L]) for a single non-trivial framed link in a triangulated knot complement with a non-taut angle structure; a mismatch in any coefficient, or a showing that the cone relation (24) forces the gl1-skein module to be zero, would refute the 3d compatibility theorem for that example.

Watch

Extended reading notes

Core claim

The central claim is that for an ideally triangulated 3-manifold with a generalized angle structure, the composition of the 3d quantum UV-IR map F_T (from the gl2-skein module to the gl1-skein module of the branched double cover) with the evaluation map ev equals the 3d quantum trace map Tr_T tensored with the identity, after applying the gl2-to-sl2 projection π. Concretely, π ∘ F_T = (Tr_T ⊗ id) ∘ ev. The same square is proved for surfaces, and for surfaces it establishes the conjecture that the 2d quantum trace map and 2d quantum UV-IR map are related in exactly this way. A corollary is that Tr_T([L]) = p_L ∘ ev ∘ F_T([L]) for any oriented framed link L, provided the intersection pairing b

Load-bearing premise

The 3d quantum UV-IR map is defined only after choosing a generalized angle structure on (Y,T), and such a structure exists only when every boundary component of Y is a torus; if no such structure exists, or if the cone skein relation (24) is inconsistent for some angle assignment, the top arrow of the compatibility square is not available and the 3d theorem has no content for that Y.

Editorial extensions

If this is right

For knot complements and other 3-manifolds with vanishing intersection pairing H1×H2→Z, every quantum-trace value can in principle be computed through the UV-IR lift: Tr_T([L]) = p_L ∘ ev ∘ F_T([L]).
The compatibility square is natural under 2-3 Pachner moves, so the two maps change coherently when the ideal triangulation is modified.
For surfaces, the previously conjectural relation between the 2d quantum trace map and the 2d quantum UV-IR map becomes a theorem, unifying two coordinate systems on skein algebras.
The stated version of the UV-IR map makes the comparison local: checking the square on triangles and face suspensions suffices, and gluing those local squares gives the global statement.
The 3d quantum trace map consequently gains an independent, geometric construction alongside its original algebraic definition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the 3d UV-IR map depends on a generalized angle structure, the compatibility is really a statement about a family of maps parametrized by Θ; varying Θ should yield identities among quantum traces and may make the trace map part of a flat family over the affine space of angle structures.
The local nature of the proof suggests a practical computational strategy: split a link into face suspensions, compute the UV-IR lifts locally, and reassemble; this could make quantum trace computations feasible in triangulations too large for direct algebraic presentations.
The gl2-to-sl2 decomposition and local-square pattern may extend to higher-rank skein modules, which the paper itself leaves as future work; a natural test would be whether the sl_n trace and gl_n UV-IR maps satisfy an analogous square with the same π and ev, now carrying n-component data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, this paper genuinely delivers: it proves the Neitzke–Yan conjecture, constructs a stated 3d quantum UV-IR map, and establishes the compatibility square with the quantum trace map. Second, the paper's own Remark 3.6 creates a real tension with Theorem C: the 3d UV-IR map is only defined after choosing a generalized angle structure, and such a structure exists only when every boundary component is a torus or Klein bottle. The stress-test example is right: a genus-2 handlebody has H2 = 0, so the intersection pairing hypothesis in Theorem C is satisfied while no angle structure exists. The theorem is not just missing a hypothesis; the statement is ill-posed for such manifolds. The intended cusped setting with torus cusps is fine, and the figure-8 example works, but the abstract's "mild hypothesis" is misleading.

What is actually new is substantial. The sign-twisted product technology, the local evaluation maps on triangles and face suspensions, the proof that the 2d UV-IR map is flip-natural, and the 3d Pachner naturality together form a coherent and largely convincing machine. The paper is honest about dependencies: it relies on the authors' to-appear [PPar] for the 3d quantum trace, and it flags its modification of SQGM in footnote 11. The local computations are explicit and the gluing argument is the right strategy.

Soft spots, in proportion. The [PPar] dependency is the biggest one; the referee should demand either that [PPar] is posted or that the needed results are summarized with proofs. The flip-naturality of the 2d UV-IR map is asserted more than demonstrated; Theorem 4.26's proof is a commutativity-by-surjectivity argument, and the explicit generator checks in Remark 4.27 are only sketched. The ζ-independence in Proposition 3.14 appears delicate, and the paper never states directly that ev ∘ F_T is independent of the full affine space of angle structures, only that one free parameter in a specific Pachner transition drops out. None of these undercuts the main narrative, but they warrant referee scrutiny.

Who is this for? People working in quantum skein theory, cluster algebras, and spectral networks. It deserves a serious referee. My own recommendation: send it out, but ask the authors to fix Theorem C's statement, clarify the angle-structure dependence, and address the [PPar] dependency before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proves a compatibility theorem between the quantum trace map and the quantum UV-IR map for ideally triangulated surfaces and 3-manifolds. For a surface, it constructs a commutative square whose bottom arrow is the Bonahon-Wong quantum trace, top arrow the Neitzke-Yan quantum UV-IR map, and right vertical arrow an evaluation map, thereby proving Conjecture 4.23 from [NY20]. For a 3-manifold, a similar square is built locally on face suspensions and glued, yielding Theorem A. Under additional hypotheses on H1 and H2, the paper derives Theorem C, recovering the 3d quantum trace from the quantum UV-IR map. The proofs use stated skein modules, explicit computations on triangles, face suspensions, and the triangular bipyramid, together with gluing and naturality checks for flips and Pachner moves.

Significance. If the main results are correct, this is a significant contribution: it gives a direct and concrete proof of the Neitzke-Yan conjecture, provides a geometric interpretation of the recently introduced 3d quantum trace map, and establishes a framework for comparing two apparently different abelianization maps in skein theory. The paper is careful and systematic: the local compatibility maps are written explicitly (Theorems 4.20 and 5.11), the gluing well-definedness is checked in Proposition 5.12 and the surrounding relative tensor product relations, and the surface and 3-manifold naturality statements are verified in Theorems 4.26 and 5.13. The worked figure-8 knot example is valuable. The main reservation is that the 3d statements, especially Theorem C, omit an essential hypothesis on the existence of a generalized angle structure, without which the quantum UV-IR map is not defined.

major comments (3)

[Theorem C, Section 5.4] The hypothesis that the intersection pairing H1(Y;Z) × H2(Y;Z) → Z vanishes is insufficient for the stated conclusion. The map F_T is defined only after equipping (Y,T) with a generalized angle structure, and by Remark 3.6 such a structure exists only if every boundary component is a torus or Klein bottle. The stated hypothesis does not imply this: for Y equal to the interior of a genus-2 handlebody, H2(Y;Z)=0 so the pairing is trivially zero, but the single boundary component has genus 2, so no ideal triangulation admits a generalized angle structure and F_T is undefined. Thus the formula Tr_T([L]) = p_L ∘ ev ∘ F_T([L]) is ill-posed for such Y. The theorem should be amended to assume that Y admits a generalized angle structure for T, equivalently in the oriented case that every boundary component is a torus, and the abstract's description of the hypothesis as 'mild' should be qualified
[Theorem A, diagrams (1) and (39); Sections 3.4 and 5] The compatibility square is stated without reference to the generalized angle structure Θ, although the top arrow F_T is defined only after choosing Θ (Definition 3.5, Theorem 3.11, Definition 3.7) and the right vertical evaluation map constructed in Theorem 5.11 also depends on the angles θ_x. As written, the square is asserted for an arbitrary ideally triangulated 3-manifold, which is not meaningful when no Θ exists. The theorem should quantify over Θ, or restrict Y to manifolds with torus/Klein boundary components. Moreover, since the space of generalized angle structures is affine of dimension t+v (Remark 3.6), the paper should state explicitly whether the square commutes for each Θ and whether the composition ev ∘ F_T is independent of Θ. Proposition 3.14 checks only independence from the free parameter ζ in the 2–3 Pachner transition, not Θ-independence in general.
[Corollaries 4.19 and 5.10] The local evaluation maps are constructed as the unique maps making the local squares commute, using the surjectivity of F on the triangle and face suspension. This is a legitimate construction, but it means the local compatibility is established by definition rather than by an independent computation. The substantive content of the paper is in showing that these locally defined maps are well-defined, glue consistently across the relative tensor product, and are compatible with flips and Pachner moves. This framing should be stated explicitly, so that readers do not over-interpret the local commutativity as an independent verification.

minor comments (4)

[Introduction] Typo: 'Morerover' should be 'Moreover'. Also 'sheer coordinates' should likely be 'shear coordinates'.
[Section 3.3, Definition 3.7] The cone skein relation (24) is introduced with coefficients q^{±θ/π}. Since these are non-integer powers of q, it would help to state explicitly that the skein module is taken over the ring R_Θ and to comment on the consistency/non-vanishing of the resulting quotient, or to point to a reference where this is established.
[Section 4.5, proof of Theorem 4.24] In the local commutative diagram after Lemma 4.25, the notation 'QΓodd e△ ⊗ QΓeven e△' is introduced without defining Γ_e△; it is later referred to as a rank-5 lattice. Please define it before first use.
[Section 6] The example uses c_B = (-1)^{-1/2}, whereas the paper fixed (c_T,c_B)=(q^{-1/2},1) before Section 2.3. The compatibility with the earlier convention is explained only at the very end; a sentence at the start of the example would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the compatibility and recovery statements are supported by explicit constructions and gluing checks, not by equations that reduce to their own inputs.

full rationale

The paper is explicit that the evaluation map is constructed locally to make the compatibility square commute. In Section 4.3, Corollary 4.19 states that if such an evaluation map exists then it is unique, and Theorem 4.20 then defines ev on generators via preimages under the surjective 2d UV-IR map followed by the triangle quantum trace. Similarly, Corollary 5.10 and Theorem 5.11 do the same for face suspensions, defining ev(S_f) using Tr(S_f). This is a standard construction of an intertwining map, not a hidden circular derivation: the mathematical content lies in checking that these local definitions are well-defined, descend to relative tensor products, and remain compatible under gluing and Pachner moves. Those checks are carried out in Proposition 5.12 and Theorem 5.13 using the defining relations (V), (L), and (G) of the quantum gluing module; they are substantive verifications rather than restatements of the conclusion. The self-citations to [PPar] supply the prior construction of the 3d quantum trace map and its local splitting behavior, but the compatibility theorem does not assume what it proves: it proves the existence and naturality of the square (1). Theorem C's recovery formula is a direct corollary of the proven commutative square; while the word 'recovered' is somewhat generous because ev itself is built from local quantum-trace data, this is a framing issue, not a circular reduction. The main caveats are correctness-level rather than circularity-level: Theorem C's hypothesis (vanishing H1 x H2 pairing) does not ensure existence of the generalized angle structure required to define the 3d UV-IR map, as noted in Remark 3.6, and the non-canonical choice of angle structure is not explicitly shown to leave ev o F_T unchanged except in the Pachner-move check. These do not make the derivation circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the stated skein calculus imported from [Lê18, CL22, PPar], the newly introduced sign-twisted products and angle-dependent cone relation, and the standard quantum-torus / cluster machinery. The genuinely new per-paper inputs are the sign-twisted product on gl2 and gl1 skein algebras (Sec. 4.2), the cone 3-term relation with angle coefficients (Def. 3.7), and the evaluation map; these are internally motivated and checked rather than externally benchmarked.

free parameters (4)

Scaling constants (c_T, c_B) = (q^{-1/2}, 1)
Normalization choice in the face suspension module and SQGM (Thm. 2.20, Def. 2.22); constrained by c_B^2 = q c_T^2 and absorbable into rescaling (Rmk. 2.21). Does not affect the compatibility theorems.
Generalized angle structure Θ = unspecified continuous assignment
The 3d quantum UV-IR map and the cone 3-term relation (24) depend on the assignment of angles θ, θ', θ'' (Defs. 3.5, 3.7). Existence restricted to manifolds with torus boundary components (Rmk. 3.6).
Free parameter ζ in compatible angle structures under Pachner move = free, independence shown
Prop. 3.14 and Fig. 29: angle structures compatible with a 2-3 Pachner move form a 1-parameter family; the paper shows the transition map does not depend on ζ.
Sign identification of opposite edge shape parameters = equal sign, ẑ_e = ẑ_{e'}
Rmk. 2.26: one must identify opposite edge cone parameters with equal sign rather than opposite signs; the opposite choice breaks 2-3 Pachner compatibility.

assumptions (6)

domain assumption Stated sl2-skein structure theorems from [PPar, Lê18, CL22]: face suspension bimodule presentation (Prop. 2.16), 3d splitting map (Thm. 2.18), 3d trace map (Thm. 2.23)
Quoted from the same authors' to-appear paper [PPar] plus [Lê18, CL22]; the compatibility theorem is built on these structural results. Location: Sec. 2.2.
standard math Przytycki's classification of gl1-skein modules [Prz98]: the α-graded part is torsion-free iff the intersection pairing (α,·) vanishes on H2(Y;Z)
Used in Sec. 5.4 to justify the projection p_L and the recovery in Theorem C.
ad hoc to paper The angle-dependent 3-term skein relation (24) near cone points is consistent and well-defined
Introduced in Def. 3.7; motivated by Lemma 3.10 (quantum torus computation) and Rmk. 3.8 (becomes the quantum dilogarithm recurrence for angles π,0,0). The 3d UV-IR map is well-defined only through this relation (Thm. 3.11).
ad hoc to paper Sign-twisted products on gl2 and gl1 skein algebras are associative graded algebra structures
Defs. 4.5 and 4.10; needed so that the gl2-to-sl2 map π is multiplicative (Prop. 4.7). The cocycle property of b(·,·) is asserted.
standard math Flip transition maps θ_{τ→τ'} for square-root quantum Teichmüller space satisfy the pentagon relation
Imported from [BW11, Hia10] in Sec. 2.1.3 to make the 2d trace map natural under flips.
domain assumption Existence of ideal triangulation and associated WKB foliation / leaf space data on Y
The branched double cover and the foliation underlie the whole UV-IR construction (Secs. 3.1, 3.2); the maps are triangulation-dependent, with naturality proven separately.

invented entities (2)

Sign-twisted product on gl2 and gl1 skein algebras
purpose: Turn the gl2-to-sl2 projection π into an algebra homomorphism and keep the UV-IR map an algebra map (Props. 4.7, 4.11).
New structure from Sec. 4.2; internally justified by the cocycle identity for b(·,·); no external falsifiable handle.
Evaluation map ev (with local avatars ev_△, ev_Sf)
purpose: Right vertical arrow of the compatibility square, mapping gl1-skeins of the branched double cover into SQGM ⊗ gl1-skein; uniquely determined on triangles and face suspensions by surjectivity of F.
Constructed in Thms. 4.20 and 5.11; a map rather than a postulated entity, but it is genuinely new data introduced by this paper and carries the angle dependence.

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Pith. "Pith review of Compatibility of quantum trace and UV-IR maps." pith.science (2026). https://pith.science/paper/CGQ2I7CQ

@misc{pith2026250909100,
  author       = {Pith},
  title        = {Pith review of: Compatibility of quantum trace and UV-IR maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGQ2I7CQ}},
  note         = {Machine review of arXiv:2509.09100}
}

abstract

This paper studies the connection between the quantum trace map -- which maps the $\mathfrak{sl}_2$-skein module to the quantum Teichm\"uller space for surfaces and to the quantum gluing module for 3-manifolds -- and the quantum UV-IR map -- which maps the $\mathfrak{gl}_2$-skein module to the $\mathfrak{gl}_1$-skein module of the branched double cover. We show that the two maps are compatible in a precise sense, and that the compatibility map is natural under changes of triangulation; for surfaces, this resolves a conjecture of Neitzke and Yan. As a corollary, under a mild hypothesis on the 3-manifold, the quantum trace map can be recovered from the quantum UV-IR map, hence providing yet another construction of the recently introduced 3d quantum trace map.

Figures

Figures reproduced from arXiv: 2509.09100 by the authors.

**Figure 1.** An example of a stated ribbon tangle; this one is in a tetrahedron (≈ B3 ) with boundary marking. Definition 2.2. Let (Y, Γ) be a boundary marked 3-manifold, and let R := Z[A ± 1 2 ,(−A2 ) ± 1 2 ] be the base ring. The stated sl2-skein module Sksl2 A (Y, Γ) is the R-module generated by the isotopy classes of unoriented, stated ribbon tangles in Y , modulo the following skein 1 In 2d (i.e. for stated skein algebras),… view at source ↗

**Figure 2.** D6 × I with the canonical boundary marking is an algebra homomorphism (in fact an embedding) σc : SkAlgsl2 A (Σ) → SkAlgsl2 A (Σ′ ), defined on stated tangles by [L] 7→ X ⃗ϵ∈{±}(c×I)∩L [L ⃗ϵ Σ′], where L ⃗ϵ Σ′ ⊂ Σ ′ × I denotes the stated tangle obtained by splitting L ⊂ Σ × I along c × I and assigning the state ϵp ∈ {±} to the two newly created boundary points corresponding to the intersection point p ∈ (c × I) ∩ L… view at source ↗

**Figure 3.** 2d splitting map. The gray surface is c × I, the surface we are cutting Σ × I along. Definition 2.5. A bad arc is any stated tangle in Σ × I with 1 component that connects two boundary components abutting the same boundary puncture in a trivial way, with boundary states − and + in the counter-clockwise order when viewed from the boundary puncture; see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (36 more)

**Figure 4.** Figure 4: + − [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Edges a, b, c (and their inverses) generate the extended triangle algebra Te. Suppose Σ is a punctured surface (without boundary) equipped with an ideal triangulation τ . For each edge e ∈ τ (1) of the ideal triangulation τ , consider the following element xˆe of the b…

**Figure 6.** Figure 6: An edge e of triangulation τ is split into two bare edges a1 and a2. Definition 2.11. The square-root quantum Teichm¨uller space (a.k.a. square-root ChekhovFock algebra), denoted SQTSτ (Σ), is the sub-quantum torus of N △∈τ (2) Te generated by {xˆe}e∈τ (1) . In other …

**Figure 7.** Figure 7: A flip on edge x algebra isomorphism between (some appropriate completions of) the square-root quantum Teichm¨uller spaces given by7 θτ→τ ′ : SQTS \τ (Σ) ∼→ SQTS \τ ′(Σ) xˆ 7→ xˆ ′−1 , yˆ 7→ yfˆ (A 1 2 xˆ ′ ) = f(A − 1 2 xˆ ′ )ˆy, zˆ 7→ zgˆ (A − 1 2 xˆ ′ ) = g(A 1 2 xˆ…

**Figure 8.** Figure 8: Pentagon relation ensures that different sequences of flips from τ to τ ′ induce the same transition map. with these transition maps; the following diagram commutes:8 SQTS \τ (Σ) SkAlgsl2 A (Σ) SQTS \τ ′(Σ) θ ∼ τ→τ′ Trτ Trτ′ . 2.2. 3d quantum trace map. In this subsect…

**Figure 9.** Figure 9: SkAlg(Ddeg v)-module structure at v ∈ V (Γ) 8Here, we are setting cT = 1. For rescaled quantum trace maps, we need to rescale the transition maps accordingly; see the previous footnote [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: Bad arcs in SkAlgsl2 A (Ddeg v) Definition 2.15 ([PPar, Def. 3.34]). The reduced stated skein module Sksl2 A (Y, Γ) is the quotient Sksl2 A (Y, Γ) := I bad,+\ Sksl2 A (Y, Γ)/Ibad,− = Sksl2 A (Y, Γ) I bad,+ Sksl2 A (Y, Γ) + Sksl2 A (Y, Γ)I bad,− , where I bad,+ denotes…

**Figure 11.** Figure 11: A vertex cone Cv, an edge cone Ce, and a face suspension Sf for the face with vertices a, b, c Consider the decomposition Y = S f∈T (2) Sf of Y into face suspensions. Note, under this decomposition, each edge cone splits into two bare edge cones. We equip each face su…

**Figure 12.** Figure 12: A face suspension with the standard boundary marking sink vertices of degree 3 and 3 source vertices of degree 2, so the reduced stated skein module Sksl2 A (Sf) has a natural T ⊗2–B ⊗3 -bimodule structure. In [PPar, Sec. 4], the structure of such bimodules is complet…

**Figure 13.** Figure 13: An edge cone Ce splits into two bare edge cones a and a ′ . Note, this is a cone over [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: 2-3 Pachner move on the triangular bipyramid 2-3 −→ [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

**Figure 15.** Figure 15: The 2-3 Pachner move turns 7 face suspensions (6 exterior + 1 interior) into 9 face suspensions (6 exterior + 3 interior). The interior faces (1 on the left and 3 on the right) are highlighted in blue. Proposition 2.24. There is an R-module homomorphism φ2→3 : SQGMT2 …

**Figure 16.** Figure 16: Triangular bipyramid BP with the standard boundary marking (all the edges of the boundary marking are oriented toward the center of each face) Sksl2 A (BP) ∼= T ⊗6 ⊗ (B op) ⊗9 Ann([∅]) , where we have one factor of T for each face and one factor of B op for each edge …

**Figure 17.** Figure 17: Branch point (red dot) and branch cuts (orange squiggly lines) for an ideal triangle Given such a branched double cover Σe → Σ, [NY20] constructed an algebra homomorphism, called the quantum UV-IR map, from the gl2 -skein algebra of Σ to the gl1 -skein algebra of the …

**Figure 18.** Figure 18: 2d WKB foliation and its orientation The WKB foliation of Σ lifts to an oriented 1-dimensional foliation on Σe: we orient the leaves of the foliation of Σe so that the orientation is away from (resp. toward) the ideal vertices in sheet 1 (resp. sheet 2). This orientat…

**Figure 19.** Figure 19: Leaf space for an ideal triangle times I 3.1.1. 2d quantum UV-IR map. Now, we are ready to review the construction of the 2d quantum UV-IR map. Let L be a framed oriented link in Σ × I. By making a small isotopy if necessary, let’s assume that L is in general position…

**Figure 20.** Figure 20: Branch locus (red) and branch cuts (orange) for an ideal tetrahedron [PITH_FULL_IMAGE:figures/full_fig_p033_20.png]

**Figure 21.** Figure 21: Cone over a torus with 4 branch points clear. On each ideal tetrahedron, the WKB foliation is given by the cone over the WKB foliation of the boundary surface, which is S 2 with 4 punctures, triangulated into 4 ideal triangles; see [PITH_FULL_IMAGE:figures/full_fig_p…

**Figure 22.** Figure 22: 3d WKB foliation The corresponding leaf space looks exactly like the branch cuts shown in [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]

**Figure 23.** Figure 23: 2d and 3d spectral networks that depends on a choice of a generalized angle structure of the ideal triangulation. We describe this gl1 -skein module in this subsection. 3.3.1. Generalized angle structures. Definition 3.5. For an ideally triangulated 3-manifold Y , a g…

**Figure 24.** Figure 24: Dihedral angles θ, θ′ , θ′′ (2) The 3 numbers θ, θ ′ , and θ ′′ add up to π, for each tetrahedron. (3) For each internal edge of the ideal triangulation, the angles add up to 2π. We will denote a generalized angle structure by Θ = {θi}i∈I , where I is some indexing se…

**Figure 25.** Figure 25: Euclidean structure on each facet of the leaf space (π − θ) + (π − θ ′ ) + (π − θ ′′) = 3π − (θ + θ ′ + θ ′′) = 2π, for any three facets of the leaf space of a tetrahedron around a fixed vertex, the sum of the three inner angles add up to 2π. This, along with the cond…

**Figure 26.** Figure 26: The 3-term relation (24) near the cone point Remark 3.8. When the angles are π, 0, 0, the 3-term relation (24) is nothing but the recurrence relation for the quantum dilogarithm Ψ in Remark 3.4. This fact is used crucially in [ELPS25] to study a non-singular version o…

**Figure 27.** Figure 27: An isotopy of a link diagram on the leaf space crossing the singular point non-trivial case to check is when there are detours, in which case the images of the left-hand [PITH_FULL_IMAGE:figures/full_fig_p037_27.png]

**Figure 28.** Figure 28: Change of leaf space under the 2-3 Pachner move Let Θ2 be a generalized angle structure on T2, and let Θ3 be a generalized angle structure on T3 compatible with Θ2; there is always a 1-dimensional space of such Θ3’s. Explicitly, when viewed from the top of [PITH_FULL…

**Figure 29.** Figure 29: Compatible generalized angle structures under the 2-3 Pachner move. Here, η, η′ , η′′ are the dihedral angles of the bottom tetrahedron which are directly below θ, θ′ , θ′′, respectively, and ζ is a free parameter. The seams of the leaf space are perpendicular to the …

**Figure 30.** Figure 30: Checking the sign relation. Here, each strand is labeled by both sheets, so to get the corresponding links in Ye2 and Ye3, take the preimage under the projection. Now, it suffices to show that ϕ2→3 respects the following relative version of the 3-term relation θ ′ θ θ…

**Figure 31.** Figure 31: An angled biangular prism (D2 × I)θ (left) and the associated leaf space (right) a b c α β γ θa θb θc [PITH_FULL_IMAGE:figures/full_fig_p046_31.png]

**Figure 32.** Figure 32: An angled triangular prism (D3×I)θa,θb,θc (left) and the associated leaf space (right) Direct calculation shows that the quantum UV-IR map on the angled biangular prism (D2 × I)θ is the algebra homomorphism given by20 21 F : SkAlggl2 q (D2) → SkAlggl1 q (Df2) = SkAlgg…

**Figure 33.** Figure 33: Left: the gl1 -web Ln1,n2,n3 ; Right: an example, L1,2,−3, is shown, which should be understood as the Weyl-ordered product of the shown tangles. the splitting map is surjective. Also, such gluing gives the inverse of the splitting map, showing that the splitting map …

**Figure 34.** Figure 34: Labeling the edge cones of a face suspension and its double cover with face suspension module variables and their lifts. is surjective. Proof. Recall from Proposition 3.20 that F is a bimodule homomorphism mapping the empty skein to the empty skein. Thus, it is enough…

**Figure 35.** Figure 35: The leaf space of a face suspension with some angles labeled. θaS and θaT are the generalized angles assigned to the edges of the tetrahedra associated to the face suspension module variables aS and aT , respectively. space is shown in [PITH_FULL_IMAGE:figures/full_f…

**Figure 36.** Figure 36: A triangulation of the figure-8 knot complement, as well as the tangle K⃗ b contained inside of it (shown in blue). The gluing of the tetrahedra is controlled by the edge markings. The faces of the tetrahedra are labeled N, S, E, and W, and edges are labeled with thei…

**Figure 37.** Figure 37: The image of K⃗ b under the splitting map. The face suspension variables associated to edge cones that will be used in later computations are labeled with the corresponding shape parameter. ϵ2 ϵ1 z ′′ z y ′′ S = q − ϵ1 2 ϵ2 −ϵ1 ϵ1 ϵ1 = q − ϵ1 2 −−−−→ z ′′ ϵ2 z−ϵ1 [∅] …

**Figure 38.** Figure 38: The gl1 -web L⃗n in Sf; each strand labeled by n ∈ Z denotes the n-colored strand (equivalent to n parallel strands), and they are all flat on the leaf space. ⊗f∈T (2) [L⃗nf ] but with matching boundary conditions along boundary markings that are glued, which is the i…

**Figure 39.** Figure 39: The boundary of a tetrahedron has been unraveled; its double cover is a torus. Vertices A, B, C, and D are lifted to the double cover as A1, B1, C1, and D1 on sheet 1 and as A2, B2, C2, and D2 on sheet 2. We also show a cycle, its image under projection to the boundar…

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Rings of sl2 (c)-characters and the kauffman bracket skein module

Doug Bullock. Rings of sl2 (c)-characters and the kauffman bracket skein module. Commentarii Mathematici Helvetici , 72(4):521--542, 1997

1997

[2] [2]

Quantum traces for representations of surface groups in SL _2( C)

Francis Bonahon and Helen Wong. Quantum traces for representations of surface groups in SL _2( C) . Geom. Topol. , 15(3):1569--1615, 2011

2011

[3] [3]

Integrating quantum groups over surfaces

David Ben-Zvi, Adrien Brochier, and David Jordan. Integrating quantum groups over surfaces. J. Topol. , 11(4):874--917, 2018

2018

[4] [4]

Francesco Costantino and Thang T. Q. L\^ e . Stated skein algebras of surfaces. J. Eur. Math. Soc. (JEMS) , 24(12):4063--4142, 2022

2022

[5] [5]

Francesco Costantino and Thang T. Q. L\^e. Stated skein modules of 3-manifolds and TQFT . J. Inst. Math. Jussieu , 24(3):663--703, 2025

2025

[6] [6]

Excision of skein categories and factorisation homology

Juliet Cooke. Excision of skein categories and factorisation homology. Adv. Math. , 414:Paper No. 108848, 51, 2023

2023

[7] [7]

An embedding of skein algebras of surfaces into localized quantum tori from D ehn- T hurston coordinates

Renaud Detcherry and Ramanujan Santharoubane. An embedding of skein algebras of surfaces into localized quantum tori from D ehn- T hurston coordinates. Geom. Topol. , 29(1):313--348, 2025

2025

[8] [8]

Skein traces from curve counting , 2025

Tobias Ekholm, Pietro Longhi, Sunghyuk Park, and Vivek Shende. Skein traces from curve counting , 2025. to appear

2025