Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Torus knots in adjoint representation and Vogel's universality

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read For every simple Lie algebra, the torus knot T[4,n] (n odd) now has one universal adjoint knot polynomial in Vogel's parameters.

desk verdict A plausible extension of Vogel's universal adjoint knot polynomials to the T[4,n] family, but the key quantum dimensions are either fitted or parked on a website, so the printed evidence for all odd n is not yet closed. read the letter →

arxiv 2506.06219 v1 pith:BF7SU7UL submitted 2025-06-06 hep-th math-phmath.GTmath.MPmath.QA

Liudmila Bishler , Andrei Mironov This is my paper · ORCID
classification hep-thmath-phmath.GTmath.MPmath.QA MSC 17B1017B3757K14
keywords Vogel'suniversalityadjointrepresentationtorusknotsHOMFLY-PTpolynomialKauffmanRosso-JonesformulaquantumdimensionsChern-Simonstheory
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

Vogel's universality promises that quantities built from the adjoint representation of any simple Lie algebra can be written once, in terms of three parameters. This paper delivers the next concrete instance: a single universal adjoint polynomial $P^{[4,n]}_{\mathrm{Adj}}(u,v,w)$ for the torus knot $T[4,n]$ with odd $n$, given in formula (41). When the Vogel parameters are specialized to any simple Lie algebra, that one expression becomes the adjoint HOMFLY-PT polynomial for the $A$-series and the adjoint Kauffman polynomial for the orthogonal and symplectic series. The formula is assembled from universal quantum dimensions and the exponential factors $T^{-2n(\cdots)}$ that encode second Casimir eigenvalues, with the two difficult inputs — the quantum dimensions of the eigenspaces $X_4$ and $I$ — obtained from plethysm identities and from three consistency conditions. A reader should care because universality of this kind replaces many separate Lie-algebra computations with one rational function, and because the same framework is positioned to handle the remaining torus links.

What carries the argument

The load-bearing object is Vogel's plane: simple Lie algebras sit at isolated points on three lines with parameters $(a,b,c)$, and the substitution $u=q^a$, $v=q^b$, $w=q^c$, $T=uvw$ converts representation-theoretic data into $q$-dependent rational functions. The calculation itself runs on the Rosso-Jones formula for torus knots, in which the invariant is a sum over the Casimir eigenspaces of the $m$-th power of the adjoint, weighted by Adams-operation coefficients that compute the plethysm $\chi_R(p_{mk})$; for $m=4$ this gives fifteen universal terms rather than the forty-nine terms of the link case. Each term in formula (41) is the product of a universal quantum dimension and a factor $T^{-2n(\cdots)}$ encoding the second Casimir eigenvalue. The two non-trivial new ingredients are $qD_{X_4}$, built from the wedge-power formulas (42)-(46), and $qD_I$, determined by the three conditions in Section 5.3 together with the previously known $T[3,4]$ universal polynomial.

What would settle it

Take the explicit rational expression for $qD_I$ from the cited companion file, substitute the Vogel parameters of a concrete algebra such as $E_6$, and compute formula (41) both ways: from the universal expression and from a direct Casimir-eigenspace sum over the honest $E_6$ representations listed in Section 4.5. Any disagreement between the two evaluations, or any parameter choice where the three Section 5.3 constraints admit more than one solution, would falsify the claim of a well-defined universal polynomial.

Watch

Extended reading notes

Core claim

Formula (41) is the universal adjoint polynomial of the torus knot $T[4,n]$ with odd $n$: a single expression in the Vogel parameters $u=q^a$, $v=q^b$, $w=q^c$, $T=uvw$ that, at the parameter values of any simple Lie algebra, becomes the adjoint invariant of that algebra — the adjoint HOMFLY-PT polynomial in the $A$-series and the adjoint Kauffman polynomial in the orthogonal and symplectic series. The formula is organized as a sum over the fifteen Casimir eigenspaces appearing in the fourth Adams power of the adjoint representation, with each term carrying a universal quantum dimension and a second-Casimir exponential $T^{-2n(\cdots)}$. The genuinely new universal inputs are the quantum dimensions of the eigenspaces $X_4$ and $I$: $qD_{X_4}$ is obtained by writing $X_4$ through wedge powers of the adjoint, and $qD_I$ is fixed by three consistency conditions, the unknot value, the pure-plethysm limit, and topological invariance $T[4,3]=T[3,4]$. The paper checks the special-polynomial property, the Alexander property, and reflection invariance, and shows explicitly how the formula reproduces the known $A$, $B/C/D$, and exceptional cases.

Load-bearing premise

The load-bearing premise is that the three conditions in Section 5.3 — the unknot value, the pure-plethysm limit, and topological invariance $T[4,3]=T[3,4]$ — single out a unique universal quantum dimension $qD_I$ (and its permutations $I',I''$); if the system has more than one solution or no solution at some Vogel parameters, formula (41) is not well defined.

Editorial extensions

If this is right

  • For every simple Lie algebra, one specialization of formula (41) gives the adjoint knot invariant of $T[4,n]$ for odd $n$: the HOMFLY-PT polynomial in the $A$-series and the Kauffman polynomial in the $B$, $C$, and $D$ series.
  • The known universal adjoint invariants now cover $T[2,n]$, $T[3,3k\pm1]$, and $T[4,n]$ with odd $n$; the same method applied to formula (1) should produce the $T[4,4n]$ torus-link invariant, whose 49-term sum reduces to the 15 universal Casimir eigenspaces.
  • The construction explains phantom (virtual) representations: terms with negative quantum dimensions are necessary in the universal formula even though they are not honest representations of a given algebra, making the universality a statement about knot theory rather than about the representation ring.
  • The polynomial satisfies non-trivial consistency checks: the unknot value, the pure-plethysm limit, the special-polynomial property, the Alexander property $uvw=1$, reflection invariance, and the equality $T[4,3]=T[3,4]$.

Reading between the lines

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

  • If the three-constraint determination of $qD_I$ is genuinely unique, a numerical check on the Vogel parameters of, say, $E_6$ would expose any missing factor: compute the left side of formula (41) using the explicit $qD_I$ from the companion file and compare it with the direct sum over the $E_6$ data listed in Section 4.5.
  • The same strategy could be exported to $T[5,n]$: only the universal decomposition of $\mathrm{Ad}^5$ and the maximal-cycle characters $\psi_P([5])$ are needed, with $T[5,2]=T[2,5]$ playing the role of the topological-invariance condition.
  • The cancellation of the non-universal representations $X_3,Z_3,K_3,L_3$ suggests that higher torus cases may simplify by analogous identities, so future universal computations may be shorter than the raw Rosso-Jones sums.
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

4 major / 5 minor

Summary. This manuscript proposes a Vogel-universal formula for the adjoint HOMFLY-PT/Kauffman invariant of the torus knots T[4,n] with odd n. Starting from the Rosso-Jones formula and the Adams operation on the fourth power of the adjoint representation, the authors write P^{[4,n]}_{Adj}(u,v,w) in Eq. (41) as a sum over universal Casimir eigenspaces with known quantum dimensions and Casimir eigenvalues. The quantum dimension qD_X4 is obtained through the plethysm formulas (44)-(46), and qD_I is fixed by imposing the unknot condition, the pure-plethysm limit, and the T[4,3]=T[3,4] identity, Eqs. (47)-(49). The paper then checks the special-polynomial, Alexander, framing, and reflection properties and claims that (41) specializes to the adjoint HOMFLY-PT and Kauffman polynomials for every simple Lie algebra.

Significance. If established, the result would be a useful extension of Vogel universality from the known T[2,n] and T[3,n] families to the T[4,n] family, and the reduction of the Adams sum to fifteen Casimir eigenspaces is a genuine simplification. The paper is commendably explicit about the input used to determine qD_I and gives concrete Adams decompositions for the A, B/D, and C series, as well as dimension data for the exceptional algebras. However, the central formula is not self-contained in the submitted version: qD_X4 and qD_I are not written out and are deferred to an external website, and the only full-invariant match used in the construction is the n=3 case, which is also one of the inputs. The claimed universality for all odd n therefore rests on an extrapolation that is plausible but not yet demonstrated in the manuscript.

major comments (4)
  1. [§5.2–§5.3, Eq. (41)] Formula (41) is the central object of the paper, but the paper does not contain the explicit expressions for qD_X4 and qD_I: qD_X4 is defined only through Eqs. (44)-(46) and then referred to the external website [38], and qD_I is likewise deferred to [38]. A reader cannot verify the claimed universal polynomial from the manuscript alone, and the published version would depend on an unversioned, unrefereed URL. Please include the closed-form expressions for qD_X4 and qD_I, or at least give complete explicit definitions in an appendix, and state that these are part of the paper rather than external material.
  2. [§5.3, Eqs. (47)-(49)] The assertion that the three conditions (47)-(49) 'unambiguously fix' qD_I is not proved. For fixed (u,v,w) these conditions form a 3×3 linear system in qD_I, qD_I', qD_I'' with Vandermonde coefficient matrix [[1,1,1],[u,v,w],[u^3,v^3,w^3]], so a solution is unique if it exists. However, the paper does not show that the resulting solution satisfies the permutation relations I'=I(v,u,w) and I''=I(w,v,u), nor that it is free of unwanted poles in the Vogel region, nor that a different choice of three probe values of n would give the same function. Please provide the explicit solution and establish these consistency properties.
  3. [§5.3 and §5.4, Eqs. (47)-(59)] The paper uses the topological invariance P^{[4,3]}_{Adj}=P^{[3,4]}_{Adj} as one of the constraints that determines qD_I, and then lists this same equality in §5.4 as a 'property' (Eq. (59)). Because qD_I is n-independent, satisfying n=0,1,3 does not test the formula for odd n≥5. This is not a valid independent check of universality for the whole family. Please provide an independent test for at least one odd n≥5, for example by comparing the A-series specialization of (41) with the explicit Rosso-Jones sum built from (10) and the Appendix, or by checking the symmetry P^{[4,5]}=P^{[5,4]} once the T[5,n] universal polynomial is available.
  4. [§5.4, Eq. (54)] The special-polynomial check (54) is evaluated at u=v=1, but many individual quantum dimensions in Table 2 contain denominators {√u} and {√v} that vanish in that limit. The paper does not explain how the cancellations are organized in (41) to yield a regular limit. Please state the limiting procedure or provide the explicit simplified expression at u=v=1, so that the check in Eq. (54) is actually verifiable from the displayed formulas.
minor comments (5)
  1. [Conclusion] The Conclusion refers to 'Table 5.1', but the table containing the quantum dimensions is numbered Table 2; please correct the cross-reference.
  2. [Throughout] There are several language slips that should be corrected: 'they are have just the same eigenvalues' in §1, 'In variance with' in §4.4, and 'celebrates a set of properties' in §5.4.
  3. [§3.1] The symbol A is overloaded: in §3.1 it denotes q^N in the uniform HOMFLY-PT polynomial, while the standard HOMFLY variable is also called A in the same paragraph. Please use a distinct notation, for example a mathsf font, to avoid confusion.
  4. [Reference [21]] Reference [21] is listed as 'Vivek Kumar Singh et al., to appear' with no arXiv identifier and no article title; please supply a citable reference or remove it.
  5. [§4.1, Eq. (27)] The symbols X3, Z3, K3, and L3 are used in Eq. (27) before their relations (28)-(29) are introduced; a short defining sentence before Eq. (27) would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

The topological-invariance 'check' (59) is an input: qD_I is defined by imposing P[4,3]=P[3,4], so formula (41) is calibrated to the checks rather than independently predicting them.

  1. fitted input called prediction [Section 5.3, Eqs. (47)-(49); Section 5.4, Eq. (59)]
    "The simplest way to obtain the formula for qD I is to use the three conditions for the universal adjoint invariant that unambiguously fix the quantum dimensions of I, I ′ and I ′′: ... The topological invariance: P [4,3] Ad j (u, v, w) = P [3,4] Ad j (u, v, w) (49). ... In fact, this property is built in, along with (47) and (48), because of the way qD I ’s are calculated."

    Equation (59) is one of the three defining constraints from which qD_I, qD_I' and qD_I'' are solved in Section 5.3. Therefore P[4,3]=P[3,4] is not an independent property of the constructed polynomial; it holds by construction for any solution of the fitted linear system. Listing it in Section 5.4 as a property/check is circular: the check cannot fail. The genuinely predictive content of formula (41) is restricted to odd n >= 5, and the paper provides no independent computation of P[4,n] for any such n. The authors are transparent that the property is 'built in', but this transparency does not remove the fact that one of the advertised consistency checks is an input to the definition of the central unknown qD_I.

full rationale

The paper is largely self-contained on the structural side: the universal Adams operation (31) follows from the decomposition of Adj^4 imported from [12], and qD_X4 is obtained from the plethysm identities (44)-(46) together with listed quantum dimensions. The circle is concentrated in the treatment of qD_I. Three conditions—unknot (47), pure plethysm (48), and topological invariance (49)—are used to 'unambiguously fix' qD_I. The first two are legitimate external constraints every adjoint torus-knot invariant must satisfy. The third, however, is a target property of the very invariant being constructed: P[4,3]=P[3,4] is derived from formula (41) only because qD_I was chosen to make it true. Thus the topological-invariance statement (59), later reported as a property, is an input rather than an output. This is partial circularity, not full equivalence: the unknot and pure-plethysm constraints are not the final prediction, and values of n >= 5 are not directly forced. Yet the central claim that 'we constructed the universal adjoint polynomial' depends on an extrapolation from n=1,0,3 to all odd n with no independent check, and the one check the paper highlights as a property is built in by construction. Hence the score is 6: one key 'prediction' reduces to a fit, while the construction retains independent content for the unverified n>=5 cases.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The main paid-for ingredient is the unknown quantum dimension of I, which is fitted to constraints, plus a black-box use of the Adj^4 decomposition from [12].

free parameters (1)
  • Quantum dimension of representation I (and I', I'') = not given in the paper; available at [38] external website
    In Section 5.3, qD_I is fixed by requiring the three conditions (47), (48), (49) to hold. This is a fit to known properties of the invariant, not a derivation from the Vogel parameters.
assumptions (3)
  • standard math Rosso-Jones formula for torus knot invariants
    Used in Eq. (2) to express the torus knot invariant as a sum over representations of R^⊗m; an established theorem in Chern-Simons/knot theory.
  • domain assumption Universal decomposition of Adj^4 into Casimir eigenspaces from [12]
    The Adams operation (31) and the whole formula (41) rely on the decomposition of the fourth power of the adjoint representation presented in [12]; the paper does not rederive it and takes it as input.
  • ad hoc to paper Uniqueness of qD_I from the three constraints (47)-(49)
    The paper asserts without proof that the unknot, pure plethysm and T[4,3]=T[3,4] conditions unambiguously fix the quantum dimension of I; no existence or uniqueness argument is given.
invented entities (1)
  • Phantom (virtual) representations (e.g., G'', I'', J, Y'_4 as formal objects with negative dimensions)
    purpose: To maintain a symmetric universal formula (31) that reduces to the correct Adams coefficients for concrete simple Lie algebras after cancellations
    Section 4.3 explicitly introduces these as 'phantom' or 'virtual' representations that are not actual representations of the particular algebra but a technical trick to describe invariants universally.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Torus knots in adjoint representation and Vogel's universality." pith.science (2026). https://pith.science/paper/BF7SU7UL

@misc{pith2026250606219,
  author       = {Pith},
  title        = {Pith review of: Torus knots in adjoint representation and Vogel's universality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BF7SU7UL}},
  note         = {Machine review of arXiv:2506.06219}
}
abstract

Vogel's universality gives a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha,\beta,\gamma$, which are homogeneous coordinates of Vogel's plane. It is associated with representation theory within the framework of Chern-Simons theory only, and gives rise to universal knot invariants. We extend the list of these latter further, and explain how to deal with the adjoint invariants for the torus knots $T[m,n]$ considering the case of $T[4,n]$ with odd $n$ in detail.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Vogel's universality and Macdonald dimensions

    hep-th 2025-07 conditional novelty 4.0 of 10

    The paper gives a single rational formula for adjoint Macdonald dimensions that unifies the simply laced Lie algebras A_n, D_n, E6, E7, E8, plus explicit mixed-root-system dimension formulas.

Reference graph

Works this paper leans on

47 extracted references · 13 canonical work pages · cited by 1 Pith paper

  1. [17]

    Mironov, R

    A. Mironov, R. Mkrtchyan, A. Morozov, JHEP02(2016) 078, arXiv:1510.05884

    arXiv 2016

  2. [1]

    Vogel,Algebraic structures on modules of diagrams, Preprint (1995), available athttps://webusers

    P. Vogel,Algebraic structures on modules of diagrams, Preprint (1995), available athttps://webusers. imj-prg.fr/~pierre.vogel/diagrams.pdf, J. Pure Appl. Algebra,215(2011) 1292-1339

    work page 1995

  3. [2]

    Vogel,The Universal Lie algebra, Preprint (1999), available athttps://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

    P. Vogel,The Universal Lie algebra, Preprint (1999), available athttps://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

    1999

  4. [3]

    Khudoteplov, A

    D. Khudoteplov, A. Morozov, A. Sleptsov,Can Yang-Baxter imply Lie algebra?, arXiv:2503.13437

    arXiv

  5. [4]

    Mkrtchyan, A.P

    R.L. Mkrtchyan, A.P. Veselov, JHEP08(2012) 153, arXiv:1203.0766

    arXiv 2012

  6. [5]

    Mkrtchyan, JHEP09(2013) 54, arXiv:1302.1507

    R.L. Mkrtchyan, JHEP09(2013) 54, arXiv:1302.1507

    arXiv 2013

  7. [6]

    Krefl, R

    D. Krefl, R. Mkrtchyan, JHEP10(2015) 45, arXiv:1506.03907 14

    arXiv 2015

  8. [7]

    Mkrtchyan, J

    R.L. Mkrtchyan, J. Geom. Phys.129(2018) 186-191, arXiv:1709.03261

    arXiv 2018

Show all 47 references
  1. [8]

    Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics,18(2001) suppl

    B. Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics,18(2001) suppl. 2003, pp. 49-82

    2001

  2. [9]

    Mkrtchyan, Nucl

    R.L. Mkrtchyan, Nucl. Phys.B921(2017) 236-249, arXiv:1610.09910

    2017 arXiv

  3. [10]

    Landsberg, L

    J.M. Landsberg, L. Manivel, Adv. Math.201(2006) 379-407

    2006

  4. [11]

    Mkrtchyan, A.N

    R.L. Mkrtchyan, A.N. Sergeev, A.P. Veselov, J. Math. Phys.53(2012) 102106, arXiv:1105.0115

    2012 arXiv

  5. [12]

    Avetisyan, A.P

    M.Y. Avetisyan, A.P. Isaev, S.O. Krivonos, R.L. Mkrtchyan, Russian Journal of Mathematical Physics,31 (2024) 379-388, arXiv:2311.05358

    2024 arXiv

  6. [13]

    Isaev, A.A

    A.P. Isaev, A.A. Provorov, Theor.Math.Phys.206(2021) 3-22, arXiv:2012.00746

    2021 arXiv

  7. [14]

    Isaev, S.O

    A.P. Isaev, S.O. Krivonos, J. Math. Phys.62(2021) 083503, arXiv:2102.08258

    2021 arXiv

  8. [15]

    Isaev, S.O

    A.P. Isaev, S.O. Krivonos, A.A. Provorov, Int. J. Mod. Phys.A38(2023) 235003, arXiv:2212.14761

    2023 arXiv

  9. [16]

    Khudaverdian, R.L

    H.M. Khudaverdian, R.L. Mkrtchyan, Lett. Math. Phys.107(2017) 1491-1514, arXiv:1602.00337

    2017 arXiv

  10. [18]

    Mironov, A

    A. Mironov, A. Morozov, Phys. Lett.B755(2016) 47-57, arXiv:1511.09077

    2016 arXiv

  11. [19]

    Khudoteplov, E

    D. Khudoteplov, E. Lanina, A. Sleptsov,Construction of Lie algebra weight system kernel via Vogel algebra, arXiv:2411.14417

    arXiv

  12. [20]

    Bishler, A

    L. Bishler, A. Mironov, A. Morozov,Macdonald deformation of Vogel’s universality and link hyperpolyno- mials, arXiv:2505.16569

    arXiv

  13. [21]

    Vivek Kumar Singh et al., to appear

  14. [22]

    Rosso, V.F.R

    M. Rosso, V.F.R. Jones, J. Knot Theory Ramifications,2(1993) 97-112

    1993

  15. [23]

    X.-S. Lin, H. Zheng, Trans.Amer.Math.Soc.362(2010) 1-18 math/0601267

    2010 arXiv

  16. [24]

    Stevan, Annales Henri Poincare11(2010) 1201-1224, arXiv:1003.2861

    S. Stevan, Annales Henri Poincare11(2010) 1201-1224, arXiv:1003.2861

    2010 arXiv

  17. [25]

    Lin Chen, Qingtao Chen, Pacific Journal of Mathematics257(2012) 267-318, arXiv:1007.1656

    2012 arXiv

  18. [26]

    Mironov, A

    A. Mironov, A. Morozov, An. Morozov, JHEP03(2012) 034, arXiv:1112.2654 [27]https://www.symmetricfunctions.com/

    2012 arXiv

  19. [28]

    Murnaghan,The Theory of Group Representations, (The Johns Hopkins Press, Baltimore, 1938)

    F.D. Murnaghan,The Theory of Group Representations, (The Johns Hopkins Press, Baltimore, 1938)

    1938

  20. [29]

    Littlewood,The Theory of Group Characters and Matrix Representations of Groups, (The Clarendon Press, Oxford, 1950

    D.E. Littlewood,The Theory of Group Characters and Matrix Representations of Groups, (The Clarendon Press, Oxford, 1950

    1950

  21. [30]

    Balantekin, P

    A.B. Balantekin, P. Cassak, J. Math. Phys.43(2002) 604-620, arXiv:hep-th/0108130

    2002 arXiv

  22. [31]

    Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    I.G. Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    1995

  23. [32]

    Cherednik, R

    I. Cherednik, R. Elliot,Refined composite invariants of torus knots via DAHA, arXiv:1503.01441

    arXiv

  24. [33]

    R. L. Mkrtchian, Phys. Lett.B105(1981) 174-176

    1981

  25. [34]

    Cvitanovic,Group Theory, Princeton University Press, Princeton, NJ, 2004

    P. Cvitanovic,Group Theory, Princeton University Press, Princeton, NJ, 2004

    2004

  26. [35]

    Mkrtchyan, A.P

    R.L. Mkrtchyan, A.P. Veselov, J. Math. Phys.52(2011) 083514, arXiv:1011.0151

    2011 arXiv

  27. [36]

    Bouchard, B

    V. Bouchard, B. Florea, M. Marino, JHEP12(2004) 035, hep-th/0405083

    2004 arXiv

  28. [37]

    M. Y. Avetisyan, R. L. Mkrtchyan,On universal quantum dimensions of certain two-parameter series of representations, arXiv:1909.02076 [38]http://wwwth.itep.ru/knotebook/knotebook/HOMFLY/universal.htm 15

    1909 arXiv

  29. [39]

    Dunin-Barkowski, A

    P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, A. Smirnov, JHEP03(2013) 021, arXiv:1106.4305

    2013 arXiv

  30. [41]

    Zhu, JHEP10(2013) 229, arXiv:1206.5886

    S. Zhu, JHEP10(2013) 229, arXiv:1206.5886

    2013 arXiv

  31. [42]

    Dunfield, S

    N.M. Dunfield, S. Gukov, J. Rasmussen, Experimental Math.15(2006) 129-159, math/0505662

    2006 arXiv

  32. [43]

    Itoyama, A

    H. Itoyama, A. Mironov, A. Morozov and A. Morozov, JHEP07(2012) 131, arXiv:1203.5978

    2012 arXiv

  33. [44]

    Mironov, A

    A. Mironov, A. Morozov and A. Morozov, AIP Conf. Proc.1562(2013) 123-155, arXiv:1306.3197

    2013 arXiv

  34. [45]

    Krefl, A

    D. Krefl, A. Schwarz, Journal of Geometry and Physics,74(2013) 119-129, arXiv:1304.7873

    2013 arXiv

  35. [46]

    Avetisyan, R.L

    M.Y. Avetisyan, R.L. Mkrtchyan, JHEP10(2021) 033, arXiv:2107.08679

    2021 arXiv

  36. [47]

    Avetisyan,Vogel’s Universality and its Applications, arXiv:2207.04302

    M. Avetisyan,Vogel’s Universality and its Applications, arXiv:2207.04302

    arXiv

  37. [48]

    Bishler, A

    L. Bishler, A. Mironov, Phys. Lett.B867(2025) 139596, arXiv:2504.13831

    2025 arXiv

  38. [49]

    Awata, H

    H. Awata, H. Kanno, A. Mironov, A. Morozov, Nucl. Phys.B949(2019) 114816, arXiv:1905.00208

    2019 arXiv

  39. [50]

    Mironov, A

    A. Mironov, A. Morozov, Nucl. Phys.B960(2020) 115191, arXiv:2003.07836 16

    2020 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.