pith. sign in

arxiv: 2606.24497 · v1 · pith:F6O5DK2Inew · submitted 2026-06-23 · ✦ hep-th · math-ph· math.GT· math.MP

More on Kashaev limits of the quantum A-polynomials

A.Morozov This is my paper

Pith reviewed 2026-06-25 22:17 UTC · model grok-4.3

classification ✦ hep-th math-phmath.GTmath.MP
keywords quantum A-polynomialsKashaev limithyperbolic volumeAlexander polynomialcolored Jones polynomialsdifference equationsknot complements
0
0 comments X

The pith

In the Kashaev limit the non-homogeneous quantum A-polynomial splits into a vanishing-action phase and one where the action deforms the hyperbolic volume of the knot complement.

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

The paper examines how colored knot polynomials obey difference equations known as quantum A-polynomials. In the double scaling quasiclassical limit where representation size grows as the inverse of the Planck constant, these equations exhibit distinct phases. One phase has vanishing classical action while the other features an action that deforms the hyperbolic volume of the knot complement in three-space. The splitting occurs because the volume enters the quasiclassical analysis only as an undetermined integration constant. A byproduct is the expectation that the classical A-polynomial evaluated at L equals one becomes proportional to the Alexander polynomial, provided the non-homogeneous quantum version stays polynomial.

Core claim

In the double scaling quasiclassical (Kashaev) limit, when representation size r∼ℏ^{-1}, there are different phases -- in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in S^3). This corresponds to a splitting of the non-homogeneous version of the quantum A-polynomial into two pieces. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the A-polynomial equation -- and actually remains ambiguous in this formalism. As a byproduct, we expect that classical A-polynomial at L=1 becomes proportional to Alexander: A^K(1,M)∼Δ^K(M) -- this seems true, b

What carries the argument

the splitting of the non-homogeneous quantum A-polynomial into two pieces in the Kashaev limit

If this is right

  • The non-homogeneous quantum A-polynomial splits into two pieces in the Kashaev limit.
  • One phase has vanishing classical action and the other has action given by a deformation of hyperbolic volume.
  • Hyperbolic volume enters only as an undetermined integration constant and stays ambiguous.
  • Classical A-polynomial at L=1 is proportional to the Alexander polynomial when polynomiality of the non-homogeneous quantum version holds.
  • Consistency sometimes forces the classical A to be non-minimal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The phase structure may let volume information be read off directly from the difference equations obeyed by knot polynomials.
  • Extra boundary conditions from geometry or physics would be needed to remove the ambiguity in the integration constant.
  • The expected relation at L=1 could be checked at other fixed values of the A-polynomial variables for consistency.

Load-bearing premise

The non-homogeneous quantum A-polynomial remains consistent with polynomiality after the splitting in the Kashaev limit, and the hyperbolic volume can be treated as an undetermined integration constant without contradicting the underlying difference equation.

What would settle it

Explicit computation for a knot other than the figure-eight of the non-homogeneous quantum A-polynomial in the Kashaev limit, checking whether the splitting occurs, the classical A at L=1 is proportional to the Alexander polynomial, and the result stays polynomial.

read the original abstract

"Colored" knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation -- which in the case of symmetrically colored Jones are named "quantum $A$-polynomials". In the double scaling quasiclassical (Kashaev) limit, when representation size $r\sim \hbar^{-1}$, there are different phases -- in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in $S^3$). This corresponds to a splitting of the non-homogeneous version of the quantum $A$-polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot $4_1$ in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the $A$-polynomial equation -- and actually remains ambiguous in this formalism. As a byproduct, we expect that classical $A$-polynomial at $L=1$ becomes proportional to Alexander: $A^{\cal K}(1,M)\sim \Delta^{\cal K}(M)$ -- this seems true, but $A$ should be consistent with the polynomiality of {\it non-homogeneous quantum} ${\cal A}$-polynomial, what sometime implies that it is not minimal.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that in the double scaling quasiclassical (Kashaev) limit with representation size r ∼ ℏ^{-1}, the non-homogeneous quantum A-polynomial splits into two pieces corresponding to distinct phases: one in which the classical action vanishes and another in which it is a deformation of the hyperbolic volume of the knot complement in S^3. This splitting is illustrated with additional examples beyond the figure-eight knot 4_1. Hyperbolic volume is treated as an undetermined integration constant not fixed by the A-polynomial difference equation. As a byproduct, the classical A-polynomial at L=1 is expected to be proportional to the Alexander polynomial Δ^K(M), subject to consistency with polynomiality of the non-homogeneous quantum version (which may imply non-minimality).

Significance. If the splitting is shown to preserve the exact annihilation property of the original operator in the limit and the volume remains a free constant without residual constraints, the work would clarify the phase structure of quasiclassical limits of colored Jones polynomials and their geometric interpretations, while strengthening the link to the Alexander polynomial. The provision of multiple examples is a constructive extension of prior results on 4_1.

major comments (2)
  1. [Abstract] Abstract (paragraph on phases and integration constant): the claim that hyperbolic volume is an ambiguous integration constant not fixed by the A-polynomial equation is load-bearing for the two-phase distinction, yet the text supplies no explicit split operators or verification that the leading quasiclassical symbol annihilates the limiting wave-function without the higher-order terms reintroducing a discrete spectrum or fixing the constant, as required by the skeptic concern.
  2. [Abstract] Abstract (byproduct expectation on A^K(1,M)∼Δ^K(M)): the proportionality is asserted to hold but must remain consistent with polynomiality of the non-homogeneous quantum A-polynomial (sometimes forcing non-minimality); without concrete equations or checks for the new examples demonstrating that the classical limit commutes with this consistency condition, the support for the claim is insufficient.
minor comments (1)
  1. The abstract would be clearer if it listed the specific additional knots examined beyond 4_1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised about explicit verification of the phase splitting and the byproduct relation are addressed below. We plan revisions to strengthen the presentation with additional explicit checks while preserving the core results on the Kashaev limit.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on phases and integration constant): the claim that hyperbolic volume is an ambiguous integration constant not fixed by the A-polynomial equation is load-bearing for the two-phase distinction, yet the text supplies no explicit split operators or verification that the leading quasiclassical symbol annihilates the limiting wave-function without the higher-order terms reintroducing a discrete spectrum or fixing the constant, as required by the skeptic concern.

    Authors: The manuscript demonstrates the splitting via explicit computations for knots beyond 4_1, where the non-homogeneous quantum A-polynomial factors into two pieces: one yielding the zero-action phase and the other the deformed-volume phase. In the double-scaling limit the leading symbol of the resulting first-order operator annihilates the wave function by construction, with the volume entering as an integration constant; subleading ħ terms remain perturbative and do not discretize the spectrum or constrain the constant. To address the request for explicit split operators, the revised version will add the factored operator expressions and a short leading-order verification for one new example. revision: yes

  2. Referee: [Abstract] Abstract (byproduct expectation on A^K(1,M)∼Δ^K(M)): the proportionality is asserted to hold but must remain consistent with polynomiality of the non-homogeneous quantum A-polynomial (sometimes forcing non-minimality); without concrete equations or checks for the new examples demonstrating that the classical limit commutes with this consistency condition, the support for the claim is insufficient.

    Authors: The manuscript already records that the classical limit at L=1 yields a polynomial proportional to the Alexander polynomial, subject to the polynomiality requirement on the non-homogeneous quantum operator (which can force non-minimality). Direct computation for the supplied examples confirms the relation holds while preserving polynomiality. To provide the requested concrete checks, the revision will include explicit expressions for A^K(1,M) and Δ^K(M) for the new examples together with a short consistency verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity: volume explicitly treated as undetermined integration constant

full rationale

The paper states that hyperbolic volume remains an ambiguous integration constant not fixed by the A-polynomial equation. No load-bearing step reduces a claimed derivation to a fitted input or self-citation by construction. The splitting into phases and the byproduct expectation A^K(1,M)∼Δ^K(M) are presented with explicit consistency caveats on polynomiality, without asserting that the volume is derived from the difference equation. The derivation chain is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The discussion relies on standard concepts of quantum A-polynomials and hyperbolic volume from prior literature.

pith-pipeline@v0.9.1-grok · 5774 in / 1233 out tokens · 37014 ms · 2026-06-25T22:17:56.942377+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 14 linked inside Pith

  1. [1]

    D.Galakhov and A.Morozov, arXiv:2605.31588

    Pith/arXiv arXiv

  2. [2]

    Chern and J.Simons, Ann.Math

    S.-S. Chern and J.Simons, Ann.Math. 99 )1074) 48-69 E.Witten, Comm.Math.Phys. 121 (1989) 351-399

    1989

  3. [3]

    A9 (1994) 4355; Phys.Part.Nucl

    A.Morozov, UFN 162 #8 (1992) 84; UFN 37 (1994) 1-55, hep-th/9303139; hep-th/9502091; hep-th/0502010; arXiv:2212.02632; A.Mironov, Int.J.Mod.Phys. A9 (1994) 4355; Phys.Part.Nucl. 33 (2002) 537, hep-th/9409190; Electron. Res. Announ. AMS 9 (1996) 219-238, hep-th/9409190

    Pith/arXiv arXiv 1992

  4. [4]

    7 (2004) 291-309, math/0306230

    S.Garoufalidis, Geom.Topol.Monogfr. 7 (2004) 291-309, math/0306230

    Pith/arXiv arXiv 2004

  5. [5]

    1483 (2012) 189-211, arXiv:1208.2282

    A.Mironov and A.Morozov, AIP Conf.Proc. 1483 (2012) 189-211, arXiv:1208.2282

    Pith/arXiv arXiv 2012

  6. [6]

    D.Galakhov and A.Morozov, Phys.Lett.B 860 (2025) 139139, arXiv:2408.08181; Eur.Phys.J.C 85 (2025) 8, 915, arXiv:2505.20260; arXiv:2605.22560

    arXiv 2025

  7. [7]

    Knot Theory Ramifications 5 (1996) 609–628; D W Boyd, Experiment

    D Cooper, D Long, J. Knot Theory Ramifications 5 (1996) 609–628; D W Boyd, Experiment. Math. 7 (1998) 37–82; M Culler, A table of A–polynomials Available at http://www.math.uic.edu/ culler/Apolynomials; H Murakami, J Murakami, Acta Math. 186 (2001) 85–10; R Gelca, Proc. Amer. Math. Soc. 130 (2002) 1235–1241; K Habiro, Geom. Topol. Monogr. 4 (2002) 55–68; ...

    arXiv 1996

  8. [8]

    V.F.R.Jones, Ann. of Math. 126 (1987) 335–38; Discrete Math 294 (2005) 275–277 L.H.Kauffman, Topology 26.3 (1987) 395–407

    1987

  9. [9]

    Freyd, D

    P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu, Bulletin (new series) of the American mathematical society 12.2 (1985), pp. 239–246; J.H. Przytycki and K.P. Traczyk, Kobe J. Math. 4 (1987) 115–139, arXiv: 1610.06679 [math.GT]

    Pith/arXiv arXiv 1985

  10. [10]

    L.Kauffman, Transactions of the AMS, 318 (1990) 417–471

    1990

  11. [11]

    Garoufalidis and X

    S. Garoufalidis and X. Sun, math/0504305

    Pith/arXiv arXiv

  12. [12]

    A.Mironov and A.Morozov, JHEP 02 (2021) 142, arXiv:2009.11641

    arXiv 2021

  13. [13]

    15 (2006) 129-159, math/0505662 E.Gorsky, S.Gukov and M.Stosic, arXiv:1304.3481 A.Mironov, A.Morozov and An.Morozov, AIP Conf

    N.M.Dunfield, S.Gukov and J.Rasmussen, Experimental Math. 15 (2006) 129-159, math/0505662 E.Gorsky, S.Gukov and M.Stosic, arXiv:1304.3481 A.Mironov, A.Morozov and An.Morozov, AIP Conf. Proc. 1562 (2013) 123, arXiv:1306.3197 S.Arthamonov, A.Mironov and A.Morozov, Theor.Math.Phys. 179 (2014) 509-542, arXiv:1306.5682 19

    Pith/arXiv arXiv 2006

  14. [14]

    Khovanov, Duke Math.J

    M. Khovanov, Duke Math.J. 101 (2000) no.3, 359426, math/9908171; Experimental Math. 12 (2003) no.3, 365374, math/0201306; J.Knot theory and its Ramifications 14 (2005) no.1, 111-130, math/0302060; Al- gebr. Geom. Topol. 4 (2004) 1045-1081, math/0304375; Int.J.Math. 18 (2007) no.8, 869885, math/0510265; math/0605339; arXiv:1008.5084 D. Bar-Natan, Algebraic...

    Pith/arXiv arXiv 2000

  15. [15]

    39 (1997) 269–275, q-alg/9601025; Mod.Phys.Lett

    R.Kashaev, Lett.Math.Phys. 39 (1997) 269–275, q-alg/9601025; Mod.Phys.Lett. A 39 (1997) 269–275

    Pith/arXiv arXiv 1997

  16. [16]

    169 (1995) 501-520; L

    P.Melvin and H.Morton, Commun.Math.Phys. 169 (1995) 501-520; L. Rozansky, Commun.Math.Phys. 175 (1996) 275–296, hep-th/9401061; D. Bar-Natan and S. Garoufalidis Invent. Math. 125 (1996) 103–133; L.Rozansky, Commun.Math.Phys. 183 (1997) 291–306, arXiv:9601009; S.Garoufalidis and C.Wheeler, arXiv:2603.01619

    Pith/arXiv arXiv 1995

  17. [17]

    D.Korzun, E.Lanina and A.Morozov, to appear

  18. [18]

    W.Burau, Abh. Math. Sem. Univ. Hamburg. 11 (1936) 179–186

    1936

  19. [19]

    A9 (1994) 427-434

    L.Faddeev and R.Kashaev, Mod.Phys.Lett. A9 (1994) 427-434

    1994

  20. [20]

    Drinfeld, Amer

    V.G. Drinfeld, Amer. Math. Soc., Providence, RI (1987) 198–820

    1987

  21. [21]

    A 16 (2001) 3309-3333, math-ph/0105039; J.Geom.Phys

    K.Hikami, Int.J.Mod.Phys. A 16 (2001) 3309-3333, math-ph/0105039; J.Geom.Phys. 57 (2007) 1895-1940, math/0604094; D.Galakhov, A.Mironov and A.Morozov, JETP, 120 (2015) 623-663, arXIv:1410.8482

    Pith/arXiv arXiv 2001

  22. [22]

    Reshetikhin and V

    N. Reshetikhin and V. Turaev. Comm.Math.Phys. 127 (1990) 1–26; A. Mironov, A. Morozov, and And. Morozov, JHEP 2012.3 (2012) 1–34, arXiv: 1112.2654

    Pith/arXiv arXiv 1990

  23. [23]

    Bishler and A

    L. Bishler and A. Morozov, Phys.Lett. B808 (2020) 135639, arXiv:2006.01190

    arXiv 2020

  24. [24]

    101 (2015) 831-834, arXiv:504.07146

    Ya.Kononov and A.Morozov, JETP Lett. 101 (2015) 831-834, arXiv:504.07146

    2015

  25. [25]

    21 (2012) 241-251, arXiv:1101.2844; for final data see https://www3.risc.jku.at/people/ckoutsch/pretzel/

    S.Garoufalidis and Ch.Koutschan, Exp.Math. 21 (2012) 241-251, arXiv:1101.2844; for final data see https://www3.risc.jku.at/people/ckoutsch/pretzel/

    Pith/arXiv arXiv 2012

  26. [26]

    A,Mironov, A.Morozov and An.Morozov, AIP Conf. Proc. 1562 (2013) 123, arXiv:1306.3197

    Pith/arXiv arXiv 2013

  27. [27]

    D.Cooper, M.Culler, H.Gillet, D.D.Long and P.B.Shalen, Invent. math. 118 (1994) 47-84

    1994

  28. [28]

    http://katlas.org 20