The empty-sector obstruction to Dehn-filling base change

FPRD Lab research note — 8 September 2026

Status

This note proves that the ordinary torus skein algebra is too small to encode Dehn filling by a universal balanced tensor product. More precisely, for every hyperbolic knot exterior, the natural empty-sector tensor map for meridional filling already has a nonzero kernel at the classical skein parameter $A=-1$.

The result rejects the first construction selected in the Dehn-filled quantum-trace hunt packet. It does not refute Panitch–Romo Conjecture 7.1, establish a generic- parameter obstruction, or construct the conjectural filled quantum trace.

The proof also identifies the correct intrinsic replacement: a coend over the full torus skein category, retaining every boundary-endpoint sector. The next attack is therefore sharply defined and does not require a census of knots, slopes, triangulations, or skein bases.

1. The proposed empty-sector formula

Let $B_A=\operatorname{Sk}_A(T^2\times[0,1])$ be the ordinary Kauffman-bracket skein algebra of the torus. If $M$ has torus boundary and $V_s$ is the solid torus attached along a slope $s$, there is always a natural gluing map

$$\Pi_A:\operatorname{Sk}_A(V_s) \underset{B_A}{\otimes} \operatorname{Sk}_A(M) \longrightarrow \operatorname{Sk}_A(M(s)).$$

The hunt proposed using the analogous tensor product on the quantum-gluing side. That proposal would be structurally compelling if $\Pi_A$ were the topological excision isomorphism it was intended to mirror. The following theorem shows that it is not.

2. Uniform obstruction at $A=-1$

Write

$$S_{-1}(X)=\operatorname{Sk}(X;\mathbb C,A=-1).$$

Under the standard Kauffman-bracket convention, an $\mathrm{SL}_2(\mathbb C)$ representation $\rho$ defines the evaluation

$$\operatorname{ev}_\rho([K])=-\operatorname{tr}\rho(K).$$

This is the convention of Przytycki–Sikora, Theorem 2.8. Only the existence of these evaluations is used below; no reducedness assertion about a character scheme is needed.

Theorem 2.1 (empty-sector gluing is not injective)

Let $M=S^3\setminus\nu K$ be the exterior of any hyperbolic knot, let $\mu$ be its meridian, and let $V_\mu$ be the solid torus used in meridional filling. Then the natural gluing map

$$\boxed{ \Pi_{-1}:S_{-1}(V_\mu) \underset{S_{-1}(T^2)}{\otimes} S_{-1}(M) \longrightarrow S_{-1}(S^3)\cong\mathbb C }$$

has a nonzero kernel.

Proof

Choose the lift of the complete hyperbolic holonomy whose meridian has trace $2$, and conjugate it so that

$$U=\rho(\mu)= \begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

Such a lift is standard; Dunfield–Friedl–Jackson discuss the normalization in Section 4.1. Every peripheral matrix commutes with $U$. Hence a preferred longitude has the form

$$\rho(\lambda)= \begin{pmatrix}\varepsilon&b\\0&\varepsilon\end{pmatrix}, \qquad \varepsilon\in\{1,-1\}.$$

For all $m,n\in\mathbb Z$,

$$\operatorname{tr}\rho(\mu^m\lambda^n)=2\varepsilon^n.$$

Define a solid-torus representation $\sigma$ by sending the core to $\varepsilon I$. Under the filling identification its boundary values are $\sigma(\mu)=I$ and $\sigma(\lambda)=\varepsilon I$. Thus $\rho$ and $\sigma$ have the same character on every boundary word, even though $\rho(\mu)\ne I$.

The two character evaluations consequently give a balanced functional

$$\Phi:S_{-1}(V_\mu) \underset{S_{-1}(T^2)}{\otimes}S_{-1}(M)\longrightarrow\mathbb C, \qquad \Phi(v\otimes a)=\operatorname{ev}_\sigma(v)\operatorname{ev}_\rho(a).$$

The complete holonomy group is non-elementary. There is therefore an element $g\in\pi_1(M)$ such that, writing

$$G=\rho(g)=\begin{pmatrix}a&b\\c&d\end{pmatrix},$$

we have $c\ne0$; otherwise the entire image would fix the parabolic fixed point of $U$. Set

$$\gamma=\mu g\mu g^{-1}.$$

Using $ad-bc=1$, direct multiplication gives

$$\operatorname{tr}(UGUG^{-1}) =2(ad-bc)-c^2 =2-c^2.$$

Represent $\gamma$ by an embedded framed knot $K_\gamma\subset M$ and put

$$x=1\otimes\bigl([K_\gamma]+2[\varnothing]\bigr).$$

Then

$$\Phi(x)=2-\operatorname{tr}\rho(\gamma)=c^2\ne0,$$

so $x$ is nonzero in the balanced tensor product. Meridional filling gives $S^3$, whose $A=-1$ character variety is the trivial character. Therefore $[K_\gamma]$ maps to $-2[\varnothing]$ and

$$\Pi_{-1}(x)=0.$$

Thus $0\ne x\in\ker\Pi_{-1}$. ∎

What the witness says

The obstruction is not caused by omitting another closed peripheral curve. All closed torus traces see only the semisimplification of the commuting boundary representation. They cannot distinguish

$$\rho(\mu)=I+N,\qquad N\ne0,\qquad N^2=0,$$

from $\rho(\mu)=I$ when paired only with upper-triangular peripheral matrices. Actual filling imposes the matrix equation $\rho(\mu)=I$. Nonperipheral group elements detect the nilpotent part, as the element $\gamma$ does above.

The two Frohman–Gelca generators correctly describe the solid torus as an ordinary empty-boundary-sector module. Tensoring only that sector against the ordinary skein module of a general complement loses the interface relations carried by boundary endpoints.

3. Exact scope of the obstruction

The theorem has three important boundaries.

1. Integral statement. It disproves the proposed natural isomorphism over any common Laurent coefficient ring for which specialization at $A=-1$ is defined and compatible: an isomorphism would remain an isomorphism after coefficient base change. Przytycki–Sikora Fact 2.7 supplies the required skein coefficient change.
2. Localization not excluded. The theorem does not rule out an identity after passing to a generic-parameter fraction field or after inverting a polynomial that vanishes at $A=-1$. The detected class could be specialization torsion. Any such localized theorem needs an independent proof.
3. Conjecture not refuted. Panitch–Romo ask for a nonzero quotient of the square-root quantum gluing module and a descended trace. Their intended nondegenerate conjecture could be true even though the ordinary boundary-algebra tensor formula is false.

At $A=-1$, changing a framing acts trivially, so the framing of $K_\gamma$ does not affect the witness. No $A=+1$ spin/sign conversion is used.

4. Why ordinary-skein excision does not supply the formula

Costantino–Lê’s stated-skein splitting theorem does not imply the displayed ordinary torus-algebra tensor product. Their decorated cobordism category requires every marked-surface component to carry a signed point on its boundary; a closed torus is not such an object. Their splitting theorem also requires each component of the splitting surface to meet the boundary marking. Puncturing and marking the torus gives a valid stated construction, but it changes both the interface algebra and the modules.

Gunningham–Jordan–Safronov give the appropriate excision statement. Let

$$\mathcal C=\operatorname{SkCat}(T^2)$$

be the full torus skein category, with objects given by labelled boundary configurations. Their skein-TFT gluing theorem yields a coend

$$\boxed{ \operatorname{Sk}(M(s))\cong \operatorname{SkMod}(V_s) \underset{\mathcal C}{\otimes} \operatorname{SkMod}(M). }$$

Equivalently, one may use internal skein modules over the internal algebra of the punctured torus and then take invariants. The ordinary torus skein algebra is only the invariant or empty sector. In general, invariants do not commute with relative tensor products. Their Proposition 4.3 reduces to an ordinary-algebra tensor product when both pieces are handlebodies; having only the solid-torus factor is not enough.

5. The categorical continuation

The correction does not end the attack. It replaces an insufficient object by a canonical one.

Let $e$ be the empty object of $\mathcal C$. The relative skein functor of the filling solid torus is cyclic, generated by its vacuum. With the appropriate handedness, write

$$F_{V_s}=h_e/\mathcal J_s,$$

where $h_e$ is the representable functor and $\mathcal J_s$ is the full categorical filling ideal. For any $\mathcal C$-module functor $\widetilde Q_{\mathcal T}$, co-Yoneda and right exactness give

$$F_{V_s}\underset{\mathcal C}{\otimes}\widetilde Q_{\mathcal T} \cong \widetilde Q_{\mathcal T}(e) \Big/ \operatorname{im}\!\left( \mathcal J_s\underset{\mathcal C}{\otimes}\widetilde Q_{\mathcal T} \right).$$

Suppose the square-root quantum gluing module $Q_{\mathcal T}=\operatorname{SQGM}_{\mathcal T}(M)$ is the empty value of such a functor,

$$\widetilde Q_{\mathcal T}(e)=Q_{\mathcal T},$$

and suppose the 3D quantum trace extends to a natural transformation

$$\operatorname{SkMod}(M)\longrightarrow\widetilde Q_{\mathcal T}.$$

Then filled descent follows formally by tensoring with $F_{V_s}$. Crucially, the quotient of $Q_{\mathcal T}$ now includes the images of $\mathcal J_s(X)$ for nonempty boundary objects $X$. Those are precisely the endpoint-sector relations absent from the failed empty-sector formula.

This is the next exact problem:

Construct or obstruct a full torus-skein-category extension $\widetilde Q_{\mathcal T}$ of the 3D quantum trace, including cups, caps, endpoint transport, stated boundary exchange, and square-root parity.

6. Audit of the remaining proposed inputs

6.1 Garoufalidis–Yu boundary linearity is a valid positive input

Garoufalidis–Yu Theorem 5.7 proves an exact peripheral intertwining formula for their quantum gluing target. This establishes the closed/empty-sector action in that model. It does not supply the missing endpoint sectors.

6.2 Transfer to Panitch–Park is not automatic

Chen–Kricker construct a surjection from the Garoufalidis–Yu target to the Panitch–Park square-root quantum gluing module and intertwine the two trace maps. An ambient quantum-torus map does not by itself show that the comparison kernel is stable under peripheral multiplication. Descending the action requires the normalizer condition

$$\rho^{GY}(b)\ker\mu\subseteq\ker\mu \qquad(b\in B),$$

or a direct Panitch–Park construction. Chen–Kricker Remark 5.26 notes that the displayed Garoufalidis–Yu relations may not exhaust the Panitch–Park relations, and injectivity is their Conjecture 5.27.

6.3 Classical nonvanishing needs a genuine target point

An evaluation of the Garoufalidis–Yu classical limit does not automatically descend through its surjection to Panitch–Park: it must annihilate the additional comparison kernel. Further, a nonzero $A=1$ fiber proves that an integral module is nonzero but does not prove that its generic fiber is nonzero. Finally, the peripheral eigenvalue equation $u_s=\pm1$ does not force a representation to extend over filling; Theorem 2.1 exhibits exactly the missing unipotent data.

6.4 The available Pachner theorem concerns a modified target

Panitch–Park Definition 2.22 uses a modified square-root quantum gluing module that identifies opposite edge square roots. Their Theorem 2.25 proves Pachner compatibility for that modified target. It cannot be silently identified with the original target compared by Chen–Kricker.

6.5 Degenerate triangulations require a scope guardrail

Garoufalidis–Yu explain that their quantum gluing module can vanish for degenerate triangulations, including a univalent-edge example. Whenever the Chen–Kricker surjection applies, a zero Garoufalidis–Yu source forces a zero Panitch–Park target. This is a qualification already present in the literature, not a new counterexample to the intended nondegenerate form of Panitch–Romo Conjecture 7.1.

7. Reproducibility

The companion scripts perform two independent exact audits.

• verify-empty-sector-obstruction.py implements polynomial arithmetic modulo $ad-bc=1$, verifies $\operatorname{tr}(UGUG^{-1})=2-c^2$, and checks the boundary-character agreement over a finite exact calibration grid.
• verify-peripheral-filling-base-change.py verifies the rank-two quantum torus product-to-sum identity, $\mathrm{SL}_2(\mathbb Z)$ covariance of the two Frohman–Gelca filling generators, and their classical factorization. These identities remain correct; they audit the empty-sector slope ideal but do not prove excision or filled descent.

8. Sources and intellectual dependencies

• Józef H. Przytycki and Adam S. Sikora, On skein algebras and $\mathrm{SL}_2(\mathbb C)$-character varieties, Fact 2.7 and Theorem 2.8: coefficient base change and the $A=-1$ trace evaluation.
• Nathan M. Dunfield, Stefan Friedl, and Nicholas Jackson, Twisted Alexander polynomials of hyperbolic knots, Section 4.1: the complete-holonomy lift with meridian trace $2$.
• Charles Frohman and Răzvan Gelca, Skein modules and the noncommutative torus, Theorem 5.3: the two generators of the solid-torus empty-sector ideal.
• Sam Gunningham, David Jordan, and Pavel Safronov, The finiteness conjecture for skein modules, Theorem 2.5, Theorem 4.1, and Proposition 4.3: skein-category excision, internal skein modules, and the limited ordinary-algebra reduction.
• Francesco Costantino and Thang T. Q. Lê, Stated skein modules of 3-manifolds and TQFT, Definitions 4.1 and 6.1 and Theorem 5.1: the marked-interface hypotheses.
• Stavros Garoufalidis and Tao Yu, A quantum trace map for 3-manifolds: the quantum gluing module and 3D trace, including the degenerate- triangulation warning.
• Stavros Garoufalidis and Tao Yu, The 3d-index of the 3d-skein module via the quantum trace map, Theorem 5.7: peripheral intertwining for their quantum gluing module.
• Qingjing Chen and Andrew Kricker, On 3d quantum trace maps, Theorem 1.3, Remark 5.26, and Conjecture 5.27: the comparison surjection and its unresolved kernel.
• Samuel Panitch and Sunghyuk Park, 3D quantum trace map, and Compatibility of quantum trace and UV–IR maps, Definition 2.22 and Theorem 2.25 in the latter: the modified target and its Pachner compatibility.
• Samuel Panitch and Mauricio Romo, On the length conjecture for twist knots, Conjecture 7.1: the open Dehn-filled quantum-trace problem. The authors state that Samuel Panitch is pursuing it.

The use of the full skein-category coend is a direct application of the Gunningham–Jordan–Safronov framework. The uniform holonomy witness and its application to reject the empty-sector base-change architecture are the mathematical contribution of this note. Novelty beyond this synthesis has not been externally assessed.

9. Research disposition

• Proved: ordinary torus-algebra base change fails integrally, already at $A=-1$, for every hyperbolic knot exterior under meridional filling.
• Preserved: the exact Garoufalidis–Yu peripheral intertwiner and the Frohman–Gelca empty-sector filling ideal.
• Withdrawn: the claimed ordinary-skein excision theorem, automatic transfer of boundary linearity to the Panitch–Park quotient, generic nonvanishing from a classical fiber, and any surgery-tail corollary.
• Open: Panitch–Romo Conjecture 7.1 in its intended nondegenerate setting.
• Next mode: ATTACK the full boundary-skein-category extension $\widetilde Q_{\mathcal T}$ and stop only if cups, caps, or endpoint exchange are incompatible with the square-root gluing relations.