Pith. sign in

REVIEW 2 major objections 5 minor 5 cited by

Statistics in 3d gravity from knots and links

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Fragmented knots and links, cut by external Wilson lines, produce wormhole amplitudes that give non-perturbative corrections to the statistics of OPE coefficients in the dual CFT.

desk verdict Fragmentation framework is a real step forward, but the amplitudes are oscillatory integrals without a uniform contour prescription, so the quantitative claims are conditional until that is fixed. read the letter →

arxiv 2508.10864 v1 pith:WCLPKJGC submitted 2025-08-14 hep-th

Jeevan Chandra This is my paper
classification hep-th MSC 81T4081T4557K1657R56
keywords 3dgravitywormholesOPEstatisticsVirasoroTQFTknotfragmentationnon-Gaussianitycrossingkernels2dCFTensemble
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

This paper tries to establish that every fragmentation of a knot or link by an external Wilson line yields a definite wormhole amplitude in three-dimensional gravity, and that these amplitudes are the non-perturbative corrections to the statistics of operator product expansion (OPE) coefficients in the dual two-dimensional CFT. Concretely, wormholes built from fragmentations of the trefoil, figure-eight, Solomon's, Whitehead, three-twist, and cinquefoil knots contribute to the variance, two-point, four-point (pillow and 6j), and six-point non-Gaussianities, with amplitudes expressed as oscillatory integrals of Virasoro 6j-symbols and fusion kernels times braiding phases. Different fragmentations of the same knot give the same contribution up to spin-dependent phases, and the results are shown to be consistent with the extended Gaussian ensemble of OPE data. If correct, the framework turns knot and link topology into a systematic generator of gravitational statistics corrections in 3d gravity.

What carries the argument

The central object is a fragmentation: an external Wilson line inserted into a knot or two-component link splits it into pieces carrying different conformal weights. The partition function of each fragmented network is evaluated using Virasoro TQFT crossing kernels — the fusion kernel, the Virasoro 6j-symbol, and braiding phases — and the gravitational wormhole amplitude is the squared modulus of the Virasoro TQFT partition function. The braiding phases and the pentagon and hexagon identities are what allow different fragmentations of the same knot to be compared and reduced to common integral expressions.

What would settle it

Take the Solomon's-knot amplitude in eq. (3.6) and evaluate the $P$-integral along two distinct contours that both asymptote to $\mathbb{R} + i\beta$ with $\beta > Q/4$ but wind differently around the fusion kernel's poles; if the squared modulus changes, the amplitude is contour-dependent and the central claim fails as stated. Alternatively, compute the six-point Hopf-link wormhole partition function numerically from the two different Virasoro TQFT expressions in eq. (5.8) and check whether the pentagon identity used to equate them holds to numerical precision.

Watch

Extended reading notes

Core claim

The central claim is that adding an external Wilson line to a knot or two-component link splits it into fragments, and the resulting networks are evaluated with Virasoro TQFT to give gravitational wormhole amplitudes of the form $Z_{\text{grav}} = |Z_V|^2$. For example, the trefoil-knot fragmentations give a Gaussian correction to the variance $$|c_{12a}|^2 \supset (-1)^{\ell_a}|C_{12a}|^2 \left|\int dP\, \rho_0(P)$e^{{3\pi i P^2}}$\begin{Bmatrix}1 & 2 & P\\1 & 2 & a\end{Bmatrix}\right|^2,$$ and analogous expressions are derived for the two-point non-Gaussianity, the pillow and 6j contractions, and several six-point non-Gaussianities. The paper argues that these wormholes are the non-perturbativ

Load-bearing premise

Every quoted amplitude is an oscillatory integral over Liouville momenta that does not converge on the real axis; the paper chooses a contour asymptoting to $\mathbb{R} + i\beta$ with $\beta > Q/4$, and if no canonical choice exists, the amplitudes are not uniquely defined.

Editorial extensions

If this is right

  • Each knot or link fragmentation gives a definite wormhole amplitude; different fragmentations of the same knot contribute to the same OPE contraction and differ only by spin-dependent phases.
  • Non-hyperbolic knots and links, such as the trefoil, Solomon's knot, and cinquefoil, whose complements do not admit well-defined Virasoro TQFT partition functions on their own, become computable once fragmented by a Wilson line.
  • The gravitational results for the pillow and 6j contractions are consistent with the extended Gaussian ensemble of OPE data that incorporates the Gaussian variance corrections from knots, checked via pentagon and hexagon identities.
  • Requiring equivalence between different Virasoro TQFT evaluations of the same fragmented network yields integral identities among crossing kernels, effectively giving three-dimensional derivations of the pentagon identity.
  • The framework extends systematically to six-point non-Gaussianities, where Hopf-link fragmentations by three Wilson lines provide the leading contributions to several new OPE statistics structures.

Reading between the lines

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

  • If the contour-ambiguity in the oscillatory integrals is resolved and a canonical contour exists, the correspondence between fragmented knots and rational tangles suggested in the paper could classify which fragmented knots contribute to the handlebody part of the sum over geometries and which give non-handlebody corrections.
  • The framework naturally extends to composite knots (granny and square knots) and to fragmentations of higher-genus handlebody knots, offering a route to non-perturbative corrections to higher moments of OPE data beyond the examples treated here.
  • The relative size of the variance corrections from the trefoil, figure-eight, three-twist, and cinquefoil fragmentations could be compared in a semiclassical limit to determine which knot topologies dominate the gravitational path integral, a comparison the paper leaves implicit.
  • The relation between fragmentations and rational tangles may provide a way to fix the integration contour by geometric monodromy data, potentially resolving the main technical limitation of the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a framework for computing non-perturbative corrections to OPE statistics in three-dimensional gravity by fragmenting knots and links with Wilson lines. For several knots and links with up to five crossings—trefoil, figure-eight, Hopf, Solomon, Whitehead, three-twist, and Cinquefoil—the author evaluates Virasoro TQFT partition functions on the fragmented diagrams and converts them, via Z_grav = |Z_V|^2, into contributions to the variance of OPE coefficients, a two-point non-Gaussianity, two four-point non-Gaussianities (the pillow and 6j contractions), and certain six-point non-Gaussianities. The central quantitative claims are integral formulas such as (1.4), (2.11), and (4.5), involving oscillatory exponentials times Virasoro fusion kernels or 6j-symbols. The paper also performs consistency checks of several amplitudes with an extended Gaussian ensemble of CFT data, using pentagon and hexagon identities.

Significance. If the amplitudes can be given a well-defined meaning, the framework is significant: it provides a systematic way to generate a class of non-perturbative gravitational contributions to OPE statistics from knot and link data, and it exhibits close relations between different fragmentations of the same knot. The algebraic manipulations are detailed, and the repeated reduction of calculations to Moore-Seiberg identities is a genuine strength. The central claim is not circular: the wormhole amplitudes are computed from knot diagrams using VTQFT, and the CFT2-ensemble checks are consistency tests rather than inputs. However, the paper currently leaves a load-bearing technical issue unresolved: the oscillatory integrals that define every quantitative prediction are not given a uniform, well-defined contour prescription. This issue affects the numerical content of the paper and must be addressed before the results can be considered reliable.

major comments (2)
  1. [Secs. 2.1.1, 3.2.1, Eqs. (2.3), (2.11), (3.6)-(3.8)] The paper correctly notes that the momentum integrals do not converge along the real axis, but it supplies an explicit contour only for the one-variable Solomon's-knot integral: asymptote to R + i beta with beta > Q/4. This prescription is not uniform. The two-variable integrals appearing in (2.11), (2.13), (C.6), and (4.50) contain a phase e^{2 pi i (P_s^2 - P_t^2)}. A common shift P_s -> P_s + i beta, P_t -> P_t + i beta produces a factor e^{-4 pi beta (P_s - P_t)} which is unbounded when P_s - P_t -> -infinity. A shift with opposite imaginary parts in the two variables must contend with the poles of the 6j-symbols/fusion kernels in the strip 0 < Im P < Q/2, and the paper gives no argument that the answer is independent of beta or of the homotopy class of the contour. Since every claimed amplitude—(1.4), (2.3), (2.6), (2.11), (4.5), (4.30), (C.6), etc.—is an oscillatory integral of thi
  2. [Secs. 4.1.2-4.4, Eq. (4.18), App. C, Eq. (C.3)] The CFT2-ensemble consistency checks are formal manipulations of the same undefined integrals. They use idempotency of the fusion kernel and the pentagon/hexagon identities to rewrite one oscillatory integral as another, e.g., equations (4.9), (4.15), (4.18), (4.28), and (4.60). These algebraic identities hold for any contour that avoids the relevant poles, so they cannot select a contour and cannot resolve the convergence ambiguity. The one independent check that could calibrate the prescription—the semiclassical volume of the three-twist knot complement, Eq. (C.3)—is explicitly deferred. Thus the statement that the gravitational results are 'consistent with the extended Gaussian ensemble' currently means only that the formal integral expressions are mutually consistent, not that a numerical value has been verified.
minor comments (5)
  1. [Sec. 3.3.1] The heading 'F ragmentations' contains an apparent typo; it should be 'Fragmentations'. Also, in Eq. (3.19) the prefactor begins 'ee^{-i pi(...)}', likely a typo for 'e^{-i pi(...)}'.
  2. [Appendix A] The heading 'The pantagon identity' should read 'The pentagon identity'.
  3. [Eqs. (4.20), (4.24), (4.47)] The notation 'dP dPd' is confusing; it should be written as 'dP dP_d' or 'dP dP_d' throughout. This occurs in several places and hampers readability.
  4. [General] The manuscript contains many unlabeled diagrams referenced inline, e.g., (2.2), (2.10), (3.3), (3.4), (4.19), (4.37). In the published version these figures should be numbered and given captions, and the text should refer to them explicitly, to allow the reader to verify the fragmentations.
  5. [Eq. (2.5)] Please check the prefactor e^{i pi (Delta a - 2 Delta 1 - 4 Delta 2)} in Eq. (2.5); it is not immediately clear how it is obtained from the preceding line, and a typo here would affect the claimed phase relation between the two trefoil fragmentations.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the wormhole amplitudes are computed from knot/link diagrams via VTQFT, and the ensemble checks are genuine consistency checks rather than constructions that presuppose the target results.

full rationale

The central quantities—e.g. Eqs. (1.4), (2.3), (2.11), (3.6), (4.5), (4.30)—are obtained by evaluating VTQFT partition functions on explicit knot/link fragmentations. No parameter is fitted to the OPE data being predicted, and the paper does not define the knot amplitudes in terms of the ensemble moments they are compared with. The consistency checks in Sec. 4 are bona fide cross-checks: the extended Gaussian ensemble supplies only the variance-type inputs, while the four-point non-Gaussianities are independently computed from wormhole topology; the matching steps use Moore–Seiberg/pentagon/hexagon identities, but the final equality is not assumed. The paper itself notes at Eq. (4.11) that the VTQFT calculation could conversely be viewed as a 3d derivation of the pentagon identity, which is an acknowledgment that the consistency check shares axioms with the computational framework rather than a hidden insertion of the target answer. The only serious concern raised by the skeptic—the lack of a uniform contour prescription for the oscillatory integrals—is a mathematical well-definedness/correctness issue, not a circularity issue: an undefined integral is not a prediction that reduces to its input. Self-citations such as [1] are used for context and for the ensemble benchmark, but the central wormhole computation is independent of those citations. Overall circularity is therefore minimal.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; all integrations are over Liouville momenta with the standard measure rho0(P), and external conformal weights are inputs. The central claim rests on the VTQFT/gravity dictionary, the Moore-Seiberg identities for Virasoro crossing kernels, admissible analytic continuations for oscillatory integrals, and the interpretation of wormhole boundaries as OPE moments. No new particles, forces, dimensions, or other entities are introduced.

assumptions (6)
  • domain assumption Virasoro TQFT gives the exact partition function of 3d gravity, and Z_grav(M) = |Z_V(M)|^2.
    Invoked in eq. (1.1) and used throughout; it is the dictionary that turns VTQFT evaluations into gravitational wormhole amplitudes.
  • standard math Moore-Seiberg consistency conditions (pentagon and hexagon identities, F and S kernel relations) hold for Virasoro crossing kernels.
    Used in every evaluation and consistency check; the paper cites [35-38,43,55] rather than proving them.
  • domain assumption All operator weights are above the black hole threshold, so the crossing kernels can be analytically continued without subtleties.
    Stated in Sec. 1.1.
  • domain assumption A suitable integration contour exists for the oscillatory momentum integrals, with the real-axis divergence regulated by a deformation into the complex plane.
    Needed for the amplitudes in Secs. 2 through 5; only the Solomon's knot example is discussed in detail (Sec. 3.2.1).
  • domain assumption The wormhole partition function with thrice-punctured sphere boundaries is interpreted as a moment of the OPE coefficients of a 2d CFT ensemble (Virasoro ETH).
    This is the holographic dictionary inherited from [1-3,27]; it underlies identifications like c11a c22a* as a two-point non-Gaussianity.
  • standard math Knot and link complements and their hyperbolicity or non-hyperbolicity classifications for prime knots and links up to five crossings.
    Used to decide which networks are directly computable with VTQFT and which require an external Wilson line to become well-defined.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Statistics in 3d gravity from knots and links." pith.science (2026). https://pith.science/paper/WCLPKJGC

@misc{pith2026250810864,
  author       = {Pith},
  title        = {Pith review of: Statistics in 3d gravity from knots and links},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCLPKJGC}},
  note         = {Machine review of arXiv:2508.10864}
}
abstract

In recent years, there has been remarkable progress in evaluating wormhole amplitudes in 3d Einstein gravity with negative cosmological constant and matching them to statistics of 2d CFT data. In this work, we compute non-perturbative Gaussian and non-Gaussian gravitational contributions to the OPE statistics using a framework that can systematically generate a class of such non-perturbative effects - \textit{Fragmentation of knots and links by Wilson lines}. We illustrate this idea by constructing multi-boundary wormholes from fragmentation diagrams of prime knots and links with upto five crossings. We discuss fragmentations of hyperbolic knots and links like the figure-eight knot, the three-twist knot and the Whitehead link; and non-hyperbolic ones like the Hopf link, the trefoil knot, the Solomon's knot and the Cinquefoil knot. Using Virasoro TQFT, we show how the partition functions on wormholes constructed from different fragmentations of the same knot or link are closely related. Using these fragmentations, we compute gravitational contributions to the variance, a two-point non-Gaussianity, two structures of four-point non-Gaussianities called the `pillow contraction' and the `$6j$-contraction', and some six-point non-Gaussianities. We also check the consistency of some of these non-Gaussianities with the extended Gaussian ensemble of OPE data that incorporates the Gaussian corrections to the variance from knots.

Figures

Figures reproduced from arXiv: 2508.10864 by the authors.

Figure 1
Figure 1. The figure above shows the two fragmentations of the right-handed trefoil knot by an [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The figure above shows the three types of fragmentations of the figure-eight knot by an [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Elastic stiffness of three-dimensional black holes and wormholes from Liouville line defects

    hep-th 2026-07 accept novelty 7.0 of 10

    Shape and mass-density deformations of thin-shell AdS3 black holes and wormholes have stiffness kernels equal to two-point functions of defect-local displacement and mass-density operators, computed from linearized Li...

  2. Additional constraints for the tensor bootstrap

    hep-th 2026-06 unverdicted novelty 6.0 of 10

    New positivity constraints from open bubbles and color matrices provide sharp bounds on unitary tensor integrals at finite N and probe deviations from Gaussian universality.

  3. The many facets of a hyperbolic tetrahedron: open and closed triangulations of 3d gravity

    hep-th 2026-04 conditional novelty 6.0 of 10

    Open Virasoro TQFT equals fixed-length/angle 3d gravity path integrals on compact regions and yields the CTV–scalar Virasoro relation via open-closed duality.

  4. On random matrix statistics of 3d gravity

    hep-th 2025-12 unverdicted novelty 6.0 of 10

    3d gravity on Σ_{g,n} × I with EOW branes equals the Virasoro minimal string random matrix model, with exact match for g=0 n=2 and inner-product formulation for negative Euler characteristic.

  5. The many facets of a hyperbolic tetrahedron: open and closed triangulations of 3d gravity

    hep-th 2026-04 unverdicted novelty 5.0 of 10

    Open Virasoro TQFT computes 3d gravity path integrals on compact regions using threshold-dependent boundary conditions and yields an open-closed duality relating Conformal Turaev-Viro theory to the diagonal sector of ...

Reference graph

Works this paper leans on

69 extracted references · 15 canonical work pages · cited by 4 Pith papers

  1. [1]

    Chandra, S

    J. Chandra, S. Collier, T. Hartman and A. Maloney, Semiclassical 3D gravity as an average of large-c CFTs, JHEP 12, 069, 2022, [arXiv:2203.06511 [hep-th]]

    arXiv 2022

  2. [2]

    Belin and J

    A. Belin and J. de Boer, Random statistics of OPE coefficients and Euclidean wormholes, Class. Quant. Grav. 38, 164001, 2021, [arXiv:2006.05499 [hep-th]]

    arXiv 2021

  3. [3]

    Collier, L

    S. Collier, L. Eberhardt and M. Zhang, Solving 3d gravity with virasoro tqft, 15, SciPost Physics, 2023

    work page 2023

  4. [4]

    Cotler and K

    J. Cotler and K. Jensen, AdS 3 gravity and random CFT, JHEP 04, 033, 2021, [arXiv:2006.08648 [hep-th]]

    arXiv 2021

  5. [5]

    Schlenker and E

    J.-M. Schlenker and E. Witten, No ensemble averaging below the black hole threshold, JHEP 07, 143, 2022, [arXiv:2202.01372 [hep-th]]

    arXiv 2022

  6. [6]

    D. L. Jafferis, L. Rozenberg and G. Wong, 3d Gravity as a random ensemble, 2024, [arXiv:2407.02649 [hep-th]]

    arXiv 2024

  7. [7]

    de Boer, D

    J. de Boer, D. Liska, B. Post and M. Sasieta, A principle of maximum ignorance for semiclassical gravity, JHEP 2024, 003, 2024, [arXiv:2311.08132 [hep-th]]

    arXiv 2024

  8. [8]

    Cotler and K

    J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3 gravity, JHEP 02, 079, 2019, [arXiv:1808.03263 [hep-th]]

    arXiv 2019

Show all 69 references
  1. [9]

    Maxfield and G

    H. Maxfield and G. J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01, 118, 2021, [arXiv:2006.11317 [hep-th]]. 72

    2021 arXiv

  2. [10]

    Eberhardt, Off-shell Partition Functions in 3d Gravity, Commun

    L. Eberhardt, Off-shell Partition Functions in 3d Gravity, Commun. Math. Phys. 405, 76, 2024, [arXiv:2204.09789 [hep-th]]

    2024 arXiv

  3. [11]

    Di Ubaldo and E

    G. Di Ubaldo and E. Perlmutter, AdS 3/RMT2 duality, JHEP 12, 179, 2023, [arXiv:2307.03707 [hep-th]]

    2023 arXiv

  4. [12]

    Collier, L

    S. Collier, L. Eberhardt, B. M¨ uhlmann and V. A. Rodriguez, The Virasoro minimal string, SciPost Phys. 16, 057, 2024, [arXiv:2309.10846 [hep-th]]

    2024 arXiv

  5. [13]

    Yan, More on torus wormholes in 3d gravity, JHEP 11, 039, 2023, [arXiv:2305.10494 [hep-th]]

    C. Yan, More on torus wormholes in 3d gravity, JHEP 11, 039, 2023, [arXiv:2305.10494 [hep-th]]

    2023 arXiv

  6. [14]

    F. M. Haehl, W. Reeves and M. Rozali, Euclidean wormholes in two-dimensional conformal field theories from quantum chaos and number theory, Phys. Rev. D 108, L101902, 2023, [arXiv:2309.02533 [hep-th]]

    2023 arXiv

  7. [15]

    F. M. Haehl, C. Marteau, W. Reeves and M. Rozali, Symmetries and spectral statistics in chaotic conformal field theories, JHEP 07, 196, 2023, [arXiv:2302.14482 [hep-th]]

    2023 arXiv

  8. [16]

    F. M. Haehl, W. Reeves and M. Rozali, Symmetries and spectral statistics in chaotic conformal field theories. Part II. Maass cusp forms and arithmetic chaos, JHEP 12, 161, 2023, [arXiv:2309.00611 [hep-th]]

    2023 arXiv

  9. [17]

    Yan, Puzzles in 3D Off-Shell Geometries via VTQFT, 2025, [arXiv:2502.16686 [hep-th]]

    C. Yan, Puzzles in 3D Off-Shell Geometries via VTQFT, 2025, [arXiv:2502.16686 [hep-th]]

    2025

  10. [18]

    Boruch, G

    J. Boruch, G. Di Ubaldo, F. M. Haehl, E. Perlmutter and M. Rozali, Modular-invariant random matrix theory and AdS 3 wormholes, 2025, [arXiv:2503.00101 [hep-th]]

    2025

  11. [19]

    de Boer, J

    J. de Boer, J. Kames-King and B. Post, Surgery and statistics in 3d gravity, 2025, [arXiv:2506.04151 [hep-th]]

    2025 arXiv

  12. [20]

    J. M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02, 053, 2004, [arXiv:hep-th/0401024]

    2004 arXiv

  13. [21]

    Chandra, T

    J. Chandra, T. Hartman and V. Meruliya, Statistics of three-dimensional black holes from Liouville line defects, 2024, [arXiv:2404.15183 [hep-th]]

    2024 arXiv

  14. [22]

    Chandra, Euclidean wormholes in holographic RG flows, JHEP 11, 096, 2024, [arXiv:2407.15630 [hep-th]]

    J. Chandra, Euclidean wormholes in holographic RG flows, JHEP 11, 096, 2024, [arXiv:2407.15630 [hep-th]]

    2024 arXiv

  15. [23]

    Chandra and T

    J. Chandra and T. Hartman, Coarse graining pure states in AdS/CFT, JHEP 10, 030, 2023, [arXiv:2206.03414 [hep-th]]

    2023 arXiv

  16. [24]

    Abajian, F

    J. Abajian, F. Aprile, R. C. Myers and P. Vieira, Correlation functions of huge operators in AdS3/CFT2: domes, doors and book pages, JHEP 03, 118, 2024, [arXiv:2307.13188 [hep-th]]. 73

    2024 arXiv

  17. [25]

    Sasieta, Wormholes from heavy operator statistics in AdS/CFT, JHEP 03, 158, 2023, [arXiv:2211.11794 [hep-th]]

    M. Sasieta, Wormholes from heavy operator statistics in AdS/CFT, JHEP 03, 158, 2023, [arXiv:2211.11794 [hep-th]]

    2023 arXiv

  18. [26]

    D. Wang, Z. Wang and Z. Wei, Wormholes with Ends of the World, 2025, [arXiv:2504.12278 [hep-th]]

    2025

  19. [27]

    Collier, L

    S. Collier, L. Eberhardt and M. Zhang, 3d gravity from Virasoro TQFT: Holography, wormholes and knots, 2024, [arXiv:2401.13900 [hep-th]]

    2024 arXiv

  20. [28]

    de Boer, D

    J. de Boer, D. Liska and B. Post, Multiboundary wormholes and OPE statistics, 2024, [arXiv:2405.13111 [hep-th]]

    2024 arXiv

  21. [29]

    Post and I

    B. Post and I. Tsiares, A non-rational Verlinde formula from Virasoro TQFT, JHEP 04, 015, 2025, [arXiv:2411.07285 [hep-th]]

    2025 arXiv

  22. [30]

    Hartman, Triangulating quantum gravity in AdS 3, 2025, [arXiv:2507.12696 [hep-th]]

    T. Hartman, Triangulating quantum gravity in AdS 3, 2025, [arXiv:2507.12696 [hep-th]]

    2025 arXiv

  23. [31]

    Collier, Y

    S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP 05, 212, 2019, [arXiv:1811.05710 [hep-th]]

    2019 arXiv

  24. [32]

    Collier, A

    S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2, JHEP 07, 074, 2020, [arXiv:1912.00222 [hep-th]]

    2020 arXiv

  25. [33]

    Belin, J

    A. Belin, J. de Boer and D. Liska, Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes, JHEP 06, 116, 2022, [arXiv:2110.14649 [hep-th]]

    2022 arXiv

  26. [34]

    Anous, A

    T. Anous, A. Belin, J. de Boer and D. Liska, OPE statistics from higher-point crossing, JHEP 06, 102, 2022, [arXiv:2112.09143 [hep-th]]

    2022 arXiv

  27. [35]

    Ponsot and J

    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, 1999, [arXiv:hep-th/9911110]

    1999 arXiv

  28. [36]

    Ponsot and J

    B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(sl(2,R)), Commun. Math. Phys. 224, 613–655, 2001, [arXiv:math/0007097]

    2001 arXiv

  29. [37]

    Teschner and G

    J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories, Lett. Math. Phys. 104, 527–551, 2014, [arXiv:1202.4698 [hep-th]]

    2014 arXiv

  30. [38]

    Teschner and G

    J. Teschner and G. S. Vartanov, Supersymmetric gauge theories, quantization of Mflat, and conformal field theory, Adv. Theor. Math. Phys. 19, 1–135, 2015, [arXiv:1302.3778 [hep-th]]

    2015 arXiv

  31. [39]

    H. L. Verlinde, Conformal Field Theory, 2- D Quantum Gravity and Quantization of Teichmuller Space, Nucl. Phys. B 337, 652–680, 1990. 74

    1990

  32. [40]

    A. B. Zamolodchikov and A. B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477, 577–605, 1996, [arXiv:hep-th/9506136]

    1996 arXiv

  33. [41]

    A. B. Zamolodchikov and A. B. Zamolodchikov, Liouville field theory on a pseudosphere, 2001, [arXiv:hep-th/0101152]

    2001 arXiv

  34. [42]

    Dorn and H

    H. Dorn and H. J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429, 375–388, 1994, [arXiv:hep-th/9403141]

    1994 arXiv

  35. [43]

    G. W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123, 177, 1989

    1989

  36. [44]

    Srednicki, Chaos and quantum thermalization, Physical Review E 50, 888–901, 1994

    M. Srednicki, Chaos and quantum thermalization, Physical Review E 50, 888–901, 1994

    1994

  37. [45]

    Chandra and T

    J. Chandra and T. Hartman, Toward random tensor networks and holographic codes in CFT, JHEP 05, 109, 2023, [arXiv:2302.02446 [hep-th]]

    2023 arXiv

  38. [46]

    Chandra, Euclidean wormholes for individual 2d CFTs, JHEP 04, 051, 2024, [arXiv:2305.07183 [hep-th]]

    J. Chandra, Euclidean wormholes for individual 2d CFTs, JHEP 04, 051, 2024, [arXiv:2305.07183 [hep-th]]

    2024 arXiv

  39. [47]

    Belin, J

    A. Belin, J. de Boer, D. L. Jafferis, P. Nayak and J. Sonner, Approximate CFTs and Random Tensor Models, 2023, [arXiv:2308.03829 [hep-th]]

    2023 arXiv

  40. [48]

    P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, 2019, [arXiv:1903.11115 [hep-th]]

    2019 arXiv

  41. [49]

    Saad, Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 2019, [arXiv:1910.10311 [hep-th]]

    P. Saad, Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 2019, [arXiv:1910.10311 [hep-th]]

    2019 arXiv

  42. [50]

    D. L. Jafferis, D. K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices, Phys. Rev. D 108, 066015, 2023, [arXiv:2209.02131 [hep-th]]

    2023 arXiv

  43. [51]

    W. P. Thurston, Three-Dimensional Geometry and Topology, vol. 1. Princeton University Press, Princeton, NJ, 1997

    1997

  44. [52]

    J. S. Purcell, Hyperbolic knot theory, 2020

    2020

  45. [53]

    R. M. Kashaev, The hyperbolic volume of knots from quantum dilogarithm, 1996

    1996

  46. [54]

    Witten, Quantum Field Theory and the Jones Polynomial, Commun

    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121, 351–399, 1989

    1989

  47. [55]

    Eberhardt, Notes on crossing transformations of Virasoro conformal blocks, 2023, [arXiv:2309.11540 [hep-th]]

    L. Eberhardt, Notes on crossing transformations of Virasoro conformal blocks, 2023, [arXiv:2309.11540 [hep-th]]. 75

    2023 arXiv

  48. [56]

    Thermal Probes of Hyperbolic Knots, talk at the conference Black Holes and Strongly Coupled Thermal Dynamics, SCGP, June 2025

    T. Hartman, “Thermal Probes of Hyperbolic Knots, talk at the conference Black Holes and Strongly Coupled Thermal Dynamics, SCGP, June 2025.”

    2025

  49. [57]

    Dimofte, S

    T. Dimofte, S. Gukov, J. Lenells and D. Zagier, Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group, Commun. Num. Theor. Phys. 3, 363–443, 2009, [arXiv:0903.2472 [hep-th]]

    2009 arXiv

  50. [58]

    Ellegaard Andersen and R

    J. Ellegaard Andersen and R. Kashaev, A TQFT from Quantum Teichm¨ uller Theory, Commun. Math. Phys. 330, 887–934, 2014, [arXiv:1109.6295 [math.QA]]

    2014 arXiv

  51. [59]

    Dimofte, Quantum Riemann Surfaces in Chern-Simons Theory, Adv

    T. Dimofte, Quantum Riemann Surfaces in Chern-Simons Theory, Adv. Theor. Math. Phys. 17, 479–599, 2013, [arXiv:1102.4847 [hep-th]]

    2013 arXiv

  52. [60]

    Dijkgraaf, H

    R. Dijkgraaf, H. Fuji and M. Manabe, The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings, Nucl. Phys. B 849, 166–211, 2011, [arXiv:1010.4542 [hep-th]]

    2011 arXiv

  53. [61]

    Kitaev, Anyons in an exactly solved model and beyond, Annals Phys

    A. Kitaev, Anyons in an exactly solved model and beyond, Annals Phys. 321, 2–111, 2006, [arXiv:cond-mat/0506438]

    2006 arXiv

  54. [62]

    Benjamin, C

    N. Benjamin, C. A. Keller, H. Ooguri and I. G. Zadeh, Narain to Narnia, Commun. Math. Phys. 390, 425–470, 2022, [arXiv:2103.15826 [hep-th]]

    2022 arXiv

  55. [63]

    Afkhami-Jeddi, H

    N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01, 130, 2021, [arXiv:2006.04839 [hep-th]]

    2021 arXiv

  56. [64]

    Maloney and E

    A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10, 187, 2020, [arXiv:2006.04855 [hep-th]]

    2020 arXiv

  57. [65]

    Maloney and E

    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02, 029, 2010, [arXiv:0712.0155 [hep-th]]

    2010 arXiv

  58. [66]

    Benjamin, S

    N. Benjamin, S. Collier and A. Maloney, Pure Gravity and Conical Defects, JHEP 09, 034, 2020, [arXiv:2004.14428 [hep-th]]

    2020 arXiv

  59. [67]

    Farb and D

    B. Farb and D. Margalit, A Primer on Mapping Class Groups. Princeton University Press, Princeton, 2012, doi:10.1515/9781400839049

    2012 doi

  60. [68]

    ISHII, K

    A. ISHII, K. KISHIMOTO, H. MORIUCHI and M. SUZUKI, A table of genus two handlebody-knots up to six crossings, Journal of Knot Theory and Its Ramifications 21, 1250035, 2012, [https://doi.org/10.1142/S0218216511009893]

    2012 doi

  61. [69]

    In progress

    A. Belin, S. Collier, L. Eberhardt, D. Liska and B. Post, “In progress.” 76

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.