FPRD hunt packet: the Dehn-filled 3D quantum trace

Mode: HUNT
Selected: 8 September 2026
Status: recognized open conjecture; no theorem or Publication is claimed

Exact question

Let $M$ be an oriented ideally triangulated 3-manifold with a chosen torus cusp, and let $M(p,q)$ be its Dehn filling along the primitive slope $p\mu+q\lambda$. Classical skein theory gives a surjection

$$q_S:\operatorname{Sk}(M)\twoheadrightarrow\operatorname{Sk}(M(p,q)),$$

whose kernel is generated by the handle-slide relations introduced by the attached solid torus. The 3D quantum trace gives

$$\operatorname{Tr}_{\mathcal T}:\operatorname{Sk}(M) \longrightarrow \operatorname{SQGM}_{\mathcal T}(M).$$

Panitch and Romo conjecture that there is a nonzero quotient $q_Q:\operatorname{SQGM}_{\mathcal T}(M)\twoheadrightarrow \operatorname{SQGM}_{\mathcal T}(M)_{p,q}$ and a descended trace

$$\operatorname{Tr}^{p,q}_{\mathcal T}:\operatorname{Sk}(M(p,q)) \longrightarrow \operatorname{SQGM}_{\mathcal T}(M)_{p,q}$$

such that

$$q_Q\circ\operatorname{Tr}_{\mathcal T} =\operatorname{Tr}^{p,q}_{\mathcal T}\circ q_S.$$

This is Conjecture 7.1 of Panitch–Romo. Their paper explicitly says that it is the subject of ongoing work by Samuel Panitch. That priority must be preserved in any later result.

Known frontier

• Panitch–Park and, independently, Garoufalidis–Yu construct a 3D quantum trace for an ideally triangulated cusped 3-manifold. The targets are versions of the square-root quantum gluing module / a left-right quotient of a triangulation quantum torus.
• Panitch–Romo prove the first-order length conjecture for an infinite family of twist knots by lifting insertions to the unfilled Whitehead-link complement. Their proof does not construct a trace intrinsic to the filled manifold; this missing naturality is exactly their Conjecture 7.1.
• Panitch–Park prove compatibility of quantum trace and UV–IR maps and establish triangulation naturality under a mild hypothesis. This is a likely route for the final naturality step, not yet a proof dependency for the proposed construction.
• Classical Kauffman-bracket skein theory supplies the Dehn-filling quotient by handle slides. It does not by itself identify the corresponding quotient inside the square-root quantum gluing module.

Canonical candidate object

Let

$$\mathcal S_\partial=\operatorname{SkAlg}(T^2)$$

be the boundary skein algebra, and let $V_{p,q}$ be the solid-torus skein module with boundary action twisted by the gluing map that sends its meridian to $p\mu+q\lambda$. The topological filling quotient is expected to be the appropriate balanced tensor product or coequalizer

$$\operatorname{Sk}(M(p,q)) \simeq \operatorname{Sk}(M)\underset{\mathcal S_\partial}{\otimes}V_{p,q},$$

with the usual sidedness and completion qualifications.

The selected attack asks whether the same base change exists on the quantum trace side. If the local 3D trace induces an exact boundary representation

$$\rho_\partial:\mathcal S_\partial\longrightarrow \operatorname{End}(\operatorname{SQGM}_{\mathcal T}(M))$$

and is $\mathcal S_\partial$-linear, define the candidate

$$\boxed{ \operatorname{SQGM}_{\mathcal T}(M)_{p,q} :=\operatorname{SQGM}_{\mathcal T}(M) \underset{\mathcal S_\partial}{\otimes}V_{p,q}.}$$

This formula is a research target, not a proved construction. Noncommutative sidedness, the square-root/even submodule, Weyl ordering, and completion may require a bimodule coequalizer rather than the displayed naive tensor product.

At the purely formal level, a quotient $q_Q$ supports a descended map exactly when

$$\operatorname{Tr}_{\mathcal T}(\ker q_S)\subseteq\ker q_Q.$$

Quotienting by the image of $\ker q_S$ is therefore formally possible, but that observation is not the conjecture: the quotient must be nonzero, canonical, compatible with the quantum-gluing structure, and natural under triangulation change. The boundary base-change formulation isolates all four requirements in one object.

Why this target is worth pursuing

The problem asks for an exact translation between a topological operation and an algebraic operation, rather than a comparison of scalar invariants. A solution would make the 3D quantum trace intrinsic to Dehn-filled manifolds and remove the need to lift every insertion to an unfilled parent. It would also connect skein-theoretic filling, quantum gluing equations, and the state integral in a functorial way.

The target is tightly structured: one boundary algebra, one filling module, and one universal coequalizer govern every manifold and every slope. It does not require a census of knots, triangulations, or skein bases.

First ATTACK

1. Boundary-linearity gate. Starting from the Panitch–Park splitting into face suspensions and the local trace, construct the left/right $\mathcal S_\partial$ action on the square-root quantum gluing module and prove that $\operatorname{Tr}_{\mathcal T}$ intertwines it. Track Weyl ordering and the even/square-root submodules exactly.
2. Filling coequalizer gate. Express the classical handle-slide kernel as the coequalizer determined by $V_{p,q}$. Prove that its quantum-trace image is generated by this single slope module rather than link-dependent relations.
3. Nontriviality gate. Specialize at the classical parameter ($A=-1$, equivalently the appropriate $q=1$ limit) and exhibit a map from the candidate quotient to functions on the filled $\mathrm{SL}_2(\mathbb C)$ character variety, or to a filled solution of the gluing equations. This must prove that the quotient is not zero.
4. Naturality gate. Use the Panitch–Park UV–IR compatibility to test invariance under the triangulation moves allowed by their theorem.

The decisive first question is Gate 1: does the face-suspension construction really retain the complete torus-boundary module action after passing to the square-root quantum gluing quotient?

Stop rule

Stop this route if boundary linearity fails, or if square-root parity and completion force a family of relations depending on individual links rather than the slope module. Do not continue by enumerating knots, Dehn-filling slopes, triangulations, handle slides, or basis elements.

Primary sources

• Samuel Panitch and Mauricio Romo, On the length conjecture for twist knots, arXiv:2609.02706v1 (2 September 2026), especially Conjecture 7.1: https://arxiv.org/html/2609.02706v1
• Samuel Panitch and Sunghyuk Park, 3d quantum trace map, arXiv:2403.12850; Algebraic & Geometric Topology 26 (2026): https://arxiv.org/html/2403.12850v1
• Stavros Garoufalidis and Tao Yu, A quantum trace map for 3-manifolds, arXiv:2403.12424: https://arxiv.org/abs/2403.12424
• Samuel Panitch and Sunghyuk Park, Compatibility of quantum trace and UV–IR maps, arXiv:2509.09100: https://arxiv.org/abs/2509.09100
• Prarit Agarwal, Dongmin Gang, Sangmin Lee, and Mauricio Romo, Quantum trace map for 3-manifolds and a length conjecture, arXiv:2203.15985: https://arxiv.org/html/2203.15985
• Józef H. Przytycki, Skein modules, arXiv:math/0611797: https://arxiv.org/abs/math/0611797

Selection record

The hunt also seriously screened spectrality of a nonhomogeneous golden-mean measure and a quantum Burnside theorem for prime-order graphs. The former did not yield a comparably sharp universal first bridge. The latter risks becoming a classification of coherent configurations because prime-order association-scheme results do not force schurity. The Dehn-filling target was selected because the boundary coequalizer gives a canonical, falsifiable structural attack.

Recommended next mode: ATTACK. Prove or disprove exact boundary linearity before computing any manifold-specific example.