The periodic-diffeomorphism case of the intertwiner volume conjecture for the Kauffman bracket skein algebra

Bonahon, Wong and Yang proposed a version of the volume conjecture for surface diffeomorphisms: for a pseudo-Anosov diffeomorphism $\varphi$ of a surface $S$ of negative Euler characteristic, the trace of a normalized intertwiner between irreducible representations of the Kauffman bracket skein algebra of $S$ at the roots of unity $q_n=e^{2\pi i/n}$ (n odd) grows at exponential rate $\mathrm{vol}_{hyp}(M_\varphi)/(4\pi)$, where $\mathrm{vol}_{hyp}(M_\varphi)$ is the hyperbolic volume of the mapping torus $M_\varphi$. Wang extended the conjecture to periodic diffeomorphisms, whose mapping tori are Seifert fibered with zero simplicial volume, predicting $\lim_{n\text{ odd}\to\infty}\frac{1}{n}\log|\mathrm{Trace}\,\Lambda^{q_n}|=0$. Wang proved the upper-bound half in general, the full conjecture for diffeomorphisms of order $2^m$ and for the once-punctured torus, and the limsup statement for prime-power orders. The general case is open: since a power of each intertwiner is the identity, its trace is a sum of roots of unity whose possible vanishing is not excluded by current number theory, and the conjecture is equivalent to nonvanishing of these traces for all sufficiently large odd $n$ — a question tied to problems of Myerson and Tao on small sums of roots of unity.

Background

The volume conjecture of Kashaev (1997) and Murakami–Murakami (2001) asserts that the asymptotics of a quantum knot invariant recover hyperbolic geometry: $\frac{2\pi}{n}\log|J_n(K;e^{2\pi i/n})|$ tends to the hyperbolic volume of the knot complement — where $J_n(K;q)$ is the $n$-th colored Jones polynomial, the quantum-group invariant of the knot $K$ associated with the $n$-dimensional irreducible representation of $U_q(sl_2)$ — and in the generalized form of Murakami–Murakami the target volume is the simplicial volume (Gromov's norm on homology, which is proportional to hyperbolic volume for hyperbolic manifolds and vanishes for Seifert fibered ones, the 3-manifolds decomposed into disjoint circle fibers). A far-reaching reimplementation of this paradigm for surface diffeomorphisms was proposed by Bonahon–Wong–Yang. The ingredients are as follows. The Kauffman bracket skein algebra $SK_{q^{1/2}}(S)$ of an oriented surface $S$ is the complex vector space spanned by framed links in $S\times[0,1]$ modulo the Kauffman bracket skein relations (the local relations expressing a crossing and a trivial loop in terms of smoothings), with multiplication given by stacking; it is a deformation of the coordinate ring of the $SL(2,\mathbb{C})$-character variety of $\pi_1(S)$, the algebraic set of conjugacy classes of representations $\pi_1(S)\to SL(2,\mathbb{C})$. At a primitive odd root of unity $q_n=e^{2\pi i/n}$, the irreducible representations of $SK_{q_n^{1/2}}(S)$ are classified by a classical shadow — a character $\gamma:\pi_1(S)\to SL(2,\mathbb{C})$ (required to lie in the smooth part of the character variety) — together with a puncture weight $p_v$ for each puncture $v$ satisfying $T_n(p_v)=-\mathrm{Trace}\,\gamma(\alpha_v)$, where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $\alpha_v$ is a loop around $v$; by the unicity theorem of Frohman–Kania-Bartoszynska–Lê this representation is unique up to isomorphism and has dimension $n^{3g+p-3}$ for genus $g$ with $p$ punctures. A diffeomorphism $\varphi$ acts on the skein algebra by $\varphi_\sharp$; if the character $\gamma$ is $\varphi$-invariant, the representations $\rho$ and $\rho\circ\varphi_\sharp$ are isomorphic, and irreducibility makes the intertwiner $\Lambda$ between them unique up to a scalar. Normalizing by $|\det\Lambda|=1$ makes $|\mathrm{Trace}\,\Lambda|$ depend only on $q$. Bonahon–Wong–Yang conjectured that for a pseudo-Anosov $\varphi$ (in the Nielsen–Thurston classification, the class of mapping classes whose mapping torus $M_\varphi=S\times[0,1]/(x,1)\sim(\varphi(x),0)$ admits a complete hyperbolic metric of finite volume), $\lim_{n\text{ odd}\to\infty}\frac{1}{n}\log|\mathrm{Trace}\,\Lambda^{q_n}|=\frac{1}{4\pi}\mathrm{vol}_{hyp}(M_\varphi)$; they proved this only for the figure-eight knot complement (the once-punctured torus bundle with monodromy $LR$). Wang observed that the natural complement is the periodic case: a periodic diffeomorphism (one whose mapping class has finite order $k$) has a Seifert fibered mapping torus with zero simplicial volume, so the simplicial-volume form of the conjecture predicts the limit $0$. Periodic diffeomorphisms fix a point of Teichmüller space (the moduli space of complete finite-area hyperbolic structures on $S$ up to isotopy), which supplies a canonical $\varphi$-invariant smooth character $\gamma_\varphi$: the associated discrete faithful representation $\pi_1(S)\to PSL(2,\mathbb{R})$ extends over $\pi_1(M_\varphi)$ and lifts to $SL(2,\mathbb{C})$, and its restriction to $\pi_1(S)$ is $\gamma_\varphi$; the puncture weights are $p_v=-(q+q^{-1})$ when $\mathrm{Trace}\,\gamma_\varphi(\alpha_v)=2$ and $p_v=2$ when $\mathrm{Trace}\,\gamma_\varphi(\alpha_v)=-2$. Two features make the periodic case sharp and number-theoretic. First, because $\varphi^k=\mathrm{Id}$ and the representation is irreducible, some power of the normalized intertwiner is scalar and can be taken to be the identity, so all eigenvalues of $\Lambda^{q_n}_{\varphi,\gamma_\varphi}$ are $k$-th roots of unity and $|\mathrm{Trace}|\le n^{3g+p-3}$; this immediately gives that the limit, if it exists, is at most $0$ (Wang's Theorem 5.5). Second, Myerson (1986) proved that any nonzero sum of $m$ (not necessarily distinct) $k$-th roots of unity has absolute value at least $m^{-k}$, so nonvanishing of the trace alone forces $\frac{1}{n}\log|\mathrm{Trace}|\to 0$: the conjecture is equivalent to nonvanishing of the traces for all sufficiently large odd $n$. The theory of Lam–Leung (2000) — vanishing sums of $k$-th roots of unity exist precisely in lengths belonging to the additive semigroup generated by the prime divisors of $k$ — settles nonvanishing when $k=2^m$ (the dimension $n^{3g+p-3}$ is odd for odd $n$), which is how Wang proved that case, and gives limsup results for odd prime powers; but for orders divisible by an odd prime the combinatorics permits vanishing, and the general question of how small nonzero sums of roots of unity can be was explicitly publicized by Tao and remains open. Explicit intertwiner computations are currently available for the closed torus (via the Frohman–Gelca embedding of its skein algebra into a quantum torus, a noncommutative deformation of the algebra of Laurent polynomials in two variables), the once-punctured torus (via the Chekhov–Fock algebra of quantum Teichmüller theory, the coordinate algebra of quantized shear coordinates for an ideal triangulation), and the four-punctured sphere (Pandey's explicit computation via the Chekhov–Fock algebra); more generally, embeddings of skein algebras of surfaces into localized quantum tori give an explicit reconstruction of the irreducible representations with prescribed classical shadow (Detcherry–Santharoubane).

Problem Statement

Prove or refute the periodic-case intertwiner volume conjecture (Wang, Conjecture 5.4). Let $S$ be an oriented surface with negative Euler characteristic and let $\varphi$ be a periodic diffeomorphism of $S$ of finite order $k$. Let $\gamma_\varphi$ be the $\varphi$-invariant smooth $SL(2,\mathbb{C})$-character constructed from a Teichmüller fixed point of $\varphi$ (lift to $SL(2,\mathbb{C})$ of the discrete faithful representation extending over $\pi_1(M_\varphi)$), and let the puncture weights be $p_v=-(q+q^{-1})$ if $\mathrm{Trace}\,\gamma_\varphi(\alpha_v)=2$ and $p_v=2$ if $\mathrm{Trace}\,\gamma_\varphi(\alpha_v)=-2$. For each odd $n$, let $\Lambda^{q_n}_{\varphi,\gamma_\varphi}$ be the intertwiner between the two isomorphic irreducible representations of $SK_{q_n^{1/2}}(S)$ with classical shadow $\gamma_\varphi$ and puncture weights $\{p_v\}$, where $q_n=e^{2\pi i/n}$ with $q_n^{1/2}=e^{\pi i/n}$, normalized by $|\det\Lambda^{q_n}_{\varphi,\gamma_\varphi}|=1$. The conjecture states

$$\lim_{n\text{ odd}\to\infty}\frac{1}{n}\log\left|\mathrm{Trace}\,\Lambda^{q_n}_{\varphi,\gamma_\varphi}\right|=0.$$

A complete answer is either a proof of this statement for every periodic diffeomorphism of every oriented surface with negative Euler characteristic, or an explicit counterexample (a surface, a periodic diffeomorphism, and the conjecture's character data) for which the limit fails to equal zero, for instance because the traces vanish along an infinite subsequence of odd $n$. The conjecture is equivalent, via Myerson's lower bound for nonzero sums of roots of unity together with the eigenvalue bound $|\mathrm{Trace}|\le n^{3g+p-3}$, to the statement that $\mathrm{Trace}\,\Lambda^{q_n}_{\varphi,\gamma_\varphi}\neq0$ for all sufficiently large odd $n$; this nonvanishing — controlling the cancellations inside the sum of $k$-th roots of unity determined by the intertwiner spectrum — is the precise open core. The cases already settled (order $2^m$ on all surfaces; the once-punctured torus in full) may be cited as known and need not be reproved, whereas the limsup statement for prime-power orders is only partial progress; the general case, in particular periodic diffeomorphisms of order divisible by an odd prime on any surface other than the once-punctured torus, remains to be decided.

References

1. Zhihao Wang. Kauffman bracket intertwiners and the volume conjecture. Algebr. Geom. Topol. 25 (2025) 2143–2177. DOI 10.2140/agt.2025.25.2143; arXiv:2212.01069. https://arxiv.org/abs/2212.01069
2. Francis Bonahon, Helen Wong, Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms I: conjecture and algebraic computations. arXiv:2112.12852 (2021). https://arxiv.org/abs/2112.12852
3. Francis Bonahon, Helen Wong, Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complement. arXiv:2203.05730 (2022). https://arxiv.org/abs/2203.05730
4. Rinat M. Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (1997) 269–275. DOI 10.1023/A:1007364912784. https://doi.org/10.1023/A:1007364912784
5. Hitoshi Murakami, Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186 (2001) 85–104. DOI 10.1007/bf02392716. https://doi.org/10.1007/bf02392716
6. Charles Frohman, Joanna Kania-Bartoszynska, Thang Lê. Unicity for representations of the Kauffman bracket skein algebra. Invent. Math. 215 (2019) 609–650. DOI 10.1007/s00222-018-0833-x. https://doi.org/10.1007/s00222-018-0833-x
7. Charles Frohman, Răzvan Gelca. Skein modules and the noncommutative torus. Trans. Amer. Math. Soc. 352 (2000) 4877–4888. DOI 10.1090/S0002-9947-00-02512-5. https://doi.org/10.1090/S0002-9947-00-02512-5
8. Gerald Myerson. How small can a sum of roots of unity be? Amer. Math. Monthly 93 (1986) 457–459. DOI 10.1080/00029890.1986.11971853. https://doi.org/10.1080/00029890.1986.11971853
9. Tsit Yuen Lam, Ka Hin Leung. On vanishing sums of roots of unity. J. Algebra 224 (2000) 91–109. DOI 10.1006/jabr.1999.8089. https://doi.org/10.1006/jabr.1999.8089
10. Terry Tao. How small can a sum of a few roots of unity be? MathOverflow question 46068, asked 2010-11-14. https://mathoverflow.net/questions/46068/how-small-can-a-sum-of-a-few-roots-of-unity-be
11. Tushar Pandey. The Bonahon-Wong-Yang volume conjecture for the four-puncture sphere. arXiv:2311.13151 (2023). https://arxiv.org/abs/2311.13151
12. Tushar Pandey, Ka Ho Wong. Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement. arXiv:2402.04483 (2024). https://arxiv.org/abs/2402.04483
13. Stavros Garoufalidis, Tao Yu. On the Bonahon–Wong–Yang invariants of pseudo-Anosov maps. arXiv:2501.00250 (2024). https://arxiv.org/abs/2501.00250
14. Stavros Garoufalidis, Tao Yu. A relation between the Baseilhac-Benedetti and the Bonahon-Liu-Wong-Yang invariants. arXiv:2601.03554 (2026). https://arxiv.org/abs/2601.03554
15. Calvin McPhail-Snyder. State integrals for the quantized $SL_2(\mathbb{C})$ Chern-Simons invariant. arXiv:2601.05136 (2026). https://arxiv.org/abs/2601.05136
16. Renaud Detcherry, Ramanujan Santharoubane. An embedding of skein algebras of surfaces into localized quantum tori from Dehn–Thurston coordinates. Geom. Topol. 29 (2025) 313–348. DOI 10.2140/gt.2025.29.313; arXiv:2205.03291. https://arxiv.org/abs/2205.03291