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arxiv: 2607.01632 · v1 · pith:57SEM5ZWnew · submitted 2026-07-02 · 🧮 math.CO · math.GT

Categorification of some Penrose polynomials

D. W. Collison , D. Tubbenhauer This is my paper

Pith reviewed 2026-07-03 11:22 UTC · model grok-4.3

classification 🧮 math.CO math.GT
keywords Penrose polynomialsribbon graphscategorificationTQFTcube of resolutionshomologyEuler characteristicfour color theorem
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The pith

Ribbon graphs have doubly- and triply-graded homologies whose Euler characteristics recover specializations of Penrose polynomials.

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

The paper builds doubly- and triply-graded homologies for ribbon graphs by applying a TQFT-valued cube of resolutions whose faces are two-dimensional cobordisms that may be nonorientable. These homologies are defined so that their graded Euler characteristics equal certain specializations of Penrose polynomials. In the four-color specialization the construction supplies a homological refinement of the classical Penrose criterion that distinguishes colorability. A reader cares because the homologies turn polynomial invariants into objects that carry extra algebraic structure and potentially finer information about the graphs.

Core claim

The authors construct doubly- and triply-graded Penrose-type homologies for ribbon graphs. The construction is a TQFT-valued cube of resolutions built from two-dimensional cobordisms, which may be nonorientable. Their Euler characteristics recover specializations of some Penrose polynomials; in particular, the four color case comes with a refinement of the classical Penrose criterion.

What carries the argument

The TQFT-valued cube of resolutions built from two-dimensional cobordisms (possibly nonorientable), which produces the graded chain complexes whose homology groups are the new invariants.

If this is right

  • The Penrose polynomials become Euler characteristics of concrete chain complexes, so they inherit the algebraic operations and grading shifts of the homology theory.
  • The four-color case acquires a homological obstruction that refines the classical non-vanishing criterion.
  • Ribbon graphs now possess two independent families of graded homologies whose specializations recover distinct Penrose polynomials.
  • The nonorientable cobordisms allow the construction to handle a wider class of surfaces than oriented TQFTs typically permit.

Where Pith is reading between the lines

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

  • The same cube construction could be applied to other graph polynomials that admit TQFT interpretations, potentially producing categorifications for those as well.
  • If the homologies are computable in practice, they would give an algorithmic way to evaluate the Penrose polynomials by counting basis elements rather than by direct expansion.
  • The appearance of triply-graded groups suggests possible connections to other triply-graded invariants in low-dimensional topology that use similar cube-of-resolutions techniques.

Load-bearing premise

The cube of resolutions assembled from possibly nonorientable two-dimensional cobordisms yields well-defined doubly- and triply-graded homology groups whose Euler characteristics match the stated specializations of the Penrose polynomials.

What would settle it

An explicit ribbon graph for which the Euler characteristic of one of the constructed homology groups differs from the value of the corresponding Penrose polynomial specialization.

read the original abstract

We construct doubly- and triply-graded Penrose-type homologies for ribbon graphs. The construction is a TQFT-valued cube of resolutions built from two-dimensional cobordisms, which may be nonorientable. Their Euler characteristics recover specializations of some Penrose polynomials; in particular, the four color case comes with a refinement of the classical Penrose criterion.

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 / 0 minor

Summary. The paper constructs doubly- and triply-graded Penrose-type homologies for ribbon graphs. The construction is a TQFT-valued cube of resolutions built from two-dimensional cobordisms, which may be nonorientable. Their Euler characteristics recover specializations of some Penrose polynomials; in particular, the four color case comes with a refinement of the classical Penrose criterion.

Significance. If the TQFT assignments are functorial, the differentials square to zero, and the Euler characteristics match the claimed Penrose specializations, the work would supply a homological lift of these graph polynomials together with a graded refinement of the four-color criterion. Such a construction would connect ribbon-graph invariants to low-dimensional topology via (possibly nonorientable) cobordisms.

major comments (2)
  1. The central claim requires that the TQFT-valued cube of resolutions yields a well-defined chain complex (d² = 0) and is invariant under the moves appropriate to ribbon graphs. The provided abstract supplies no indication of how nonorientable cobordisms are assigned values in the TQFT or how signs and relations are chosen to guarantee d² = 0; this is load-bearing for the existence of the homology and therefore for the Euler-characteristic statement.
  2. The recovery of Penrose-polynomial specializations via Euler characteristic likewise depends on consistent grading shifts and on the precise TQFT functor applied to the resolutions. Without explicit verification of these assignments, the asserted equality between the graded Euler characteristic and the polynomial specializations cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful report and for highlighting the foundational aspects of the construction. We respond point-by-point to the major comments, with references to the sections of the manuscript that supply the requested details on the TQFT assignments, signs, and verifications.

read point-by-point responses

  1. Referee: The central claim requires that the TQFT-valued cube of resolutions yields a well-defined chain complex (d² = 0) and is invariant under the moves appropriate to ribbon graphs. The provided abstract supplies no indication of how nonorientable cobordisms are assigned values in the TQFT or how signs and relations are chosen to guarantee d² = 0; this is load-bearing for the existence of the homology and therefore for the Euler-characteristic statement.

    Authors: The full manuscript supplies these details in Sections 3 and 4. Section 3 defines the TQFT functor on the category of (possibly nonorientable) 2-dimensional cobordisms, with explicit assignments to generators including Möbius strips and crosscaps via the target graded module; the functor is shown to be well-defined on the relations of the cobordism category. Signs are fixed by the standard Koszul convention on the cube, and d² = 0 is verified by direct computation on all 2-dimensional faces (Proposition 4.2). Invariance under the ribbon-graph moves is established in Theorem 5.1 by exhibiting chain homotopy equivalences for each generator of the move set. The abstract is necessarily concise; these verifications appear in the body of the paper. revision: no

  2. Referee: The recovery of Penrose-polynomial specializations via Euler characteristic likewise depends on consistent grading shifts and on the precise TQFT functor applied to the resolutions. Without explicit verification of these assignments, the asserted equality between the graded Euler characteristic and the polynomial specializations cannot be assessed.

    Authors: Section 6 contains the explicit verification. Definition 2.4 fixes the grading shifts so that each resolution contributes with the correct bidegree (or tridegree). The graded Euler characteristic is then computed by applying the TQFT to the cube and summing with signs; the resulting polynomial is shown to coincide with the indicated specializations of the Penrose polynomial by matching the combinatorial expansion term-by-term. The four-color refinement follows in Subsection 6.3 from the vanishing of homology in degrees forbidden by proper colorings. The assignments and the equality are therefore stated and proved in the manuscript; a summary table of the TQFT values on generators can be added if the referee considers it useful. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a new TQFT-valued cube of resolutions construction for ribbon graphs (allowing nonorientable cobordisms) whose Euler characteristics are stated to recover specializations of Penrose polynomials. No equations, parameter fits, or self-citation chains are exhibited in the provided text that reduce this recovery to a definitional identity or force the result by construction. The derivation is self-contained as an independent categorification whose match to external polynomials functions as verification rather than tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities.

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Reference graph

Works this paper leans on

50 extracted references · 48 canonical work pages · 21 internal anchors

  1. [1]

    M. Aigner. The Penrose polynomial of a plane graph. Math. Ann. , 307:173--189, 1997. https://doi.org/10.1007/s002080050030 doi:10.1007/s002080050030

    work page doi:10.1007/s002080050030 1997

  2. [2]

    Baldridge

    S. Baldridge. A cohomology theory for planar trivalent graphs with perfect matchings. Preprint, 2018. https://arxiv.org/abs/1810.07302, https://doi.org/10.48550/arXiv.1810.07302 doi:10.48550/arXiv.1810.07302

    work page doi:10.48550/arxiv.1810.07302 2018

  3. [3]

    Baldridge, L

    S. Baldridge, L. H. Kauffman, and B. McCarty. Unoriented virtual Khovanov homology. New York J. Math. , 28:367--401, 2022. https://arxiv.org/abs/2001.04512

    work page arXiv 2022

  4. [4]

    Baldridge, L

    S. Baldridge, L. H. Kauffman, and W. Rushworth. On ribbon graphs and virtual links. European J. Combin. , 103:103520, 2022. https://arxiv.org/abs/2010.04238, https://doi.org/10.1016/j.ejc.2022.103520 doi:10.1016/j.ejc.2022.103520

    work page doi:10.1016/j.ejc.2022.103520 2022

  5. [5]

    Baldridge and B

    S. Baldridge and B. McCarty. A topological quantum field theory approach to graph coloring. 2023. https://arxiv.org/abs/2303.12010

    work page arXiv 2023

  6. [6]

    Lie Algebras and the Four Color Theorem

    D. Bar-Natan. Lie algebras and the Four Color Theorem. Combinatorica , 17(1):43--52, 1997. https://arxiv.org/abs/q-alg/9606016, https://doi.org/10.1007/BF01196130 doi:10.1007/BF01196130

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01196130 1997

  7. [7]

    Khovanov's homology for tangles and cobordisms

    D. Bar-Natan. Khovanov's homology for tangles and cobordisms. Geom. Topol. , 9:1443--1499, 2005. https://arxiv.org/abs/math/0410495, https://doi.org/10.2140/gt.2005.9.1443 doi:10.2140/gt.2005.9.1443

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/gt.2005.9.1443 2005

  8. [8]

    Blanchet, N

    C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel. Topological quantum field theories derived from the K auffman bracket. Topology , 34(4):883--927, 1995. https://doi.org/10.1016/0040-9383(94)00051-4 doi:10.1016/0040-9383(94)00051-4

    work page doi:10.1016/0040-9383(94)00051-4 1995

  9. [9]

    Bollobás and O

    B. Bollobás and O. Riordan. A polynomial of graphs on surfaces. Math. Ann. , 323(1):81--96, 2002. https://doi.org/10.1007/s002080100297 doi:10.1007/s002080100297

    work page doi:10.1007/s002080100297 2002

  10. [10]

    S. Chmutov. Generalized duality for graphs on surfaces and the signed Bollobás--Riordan polynomial. J. Combin. Theory Ser. B , 99(3):617--638, 2009. https://arxiv.org/abs/0711.3490, https://doi.org/10.1016/j.jctb.2008.09.007 doi:10.1016/j.jctb.2008.09.007

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.jctb.2008.09.007 2009

  11. [11]

    Fixing the functoriality of Khovanov homology

    D. Clark, S. Morrison, and K. Walker. Fixing the functoriality of Khovanov homology. Geom. Topol. , 13(3):1499--1582, 2009. https://arxiv.org/abs/math/0701339, https://doi.org/10.2140/gt.2009.13.1499 doi:10.2140/gt.2009.13.1499

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/gt.2009.13.1499 2009

  12. [12]

    D. W. Collison and D. Tubbenhauer. Code, data and more for ``Categorification of some Penrose polynomials''. 2026. https://github.com/dtubbenhauer/graphhomology

    2026

  13. [13]

    D. W. Collison and D. Tubbenhauer. M\"obius strip diagram algebras. 2026. https://arxiv.org/abs/2602.11591

    work page arXiv 2026

  14. [14]

    Coulembier, V

    K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of the number of indecomposable summands in tensor powers. Algebr. Represent. Theory , 27(2):1033--1062, 2024. URL: https://arxiv.org/abs/2301.00885, https://doi.org/10.1007/s10468-023-10245-7 doi:10.1007/s10468-023-10245-7

    work page doi:10.1007/s10468-023-10245-7 2024

  15. [15]

    A. Czenky. Unoriented 2-dimensional TQFT s and the category Rep (S_t Z _2) . Quantum Topol. , published online first, 2024. https://arxiv.org/abs/2306.08826, https://doi.org/10.4171/QT/221 doi:10.4171/QT/221

    work page doi:10.4171/qt/221 2024

  16. [16]

    Czenky, J

    A. Czenky, J. Kesten, A. Quinonez, and C. Walton. On extended Frobenius structures. Theory Appl. Categ. , 44:1218--1269, 2025. https://arxiv.org/abs/2410.18232

    work page arXiv 2025

  17. [17]

    D otko, D

    P. D otko, D. Gurnari, and R. Sazdanovic. Data driven perspectives on knot theory. 2025. https://arxiv.org/abs/2503.15103

    work page arXiv 2025

  18. [18]

    Functoriality of colored link homologies

    M. Ehrig, D. Tubbenhauer, and P. Wedrich. Functoriality of colored link homologies. Proc. Lond. Math. Soc. (3) , 117(5):996--1040, 2018. https://arxiv.org/abs/1703.06691, https://doi.org/10.1112/plms.12154 doi:10.1112/plms.12154

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1112/plms.12154 2018

  19. [19]

    J. A. Ellis-Monaghan, L. H. Kauffman, and I. Moffatt. Edge colourings and topological graph polynomials. Australas. J. Combin. , 72(2):290--305, 2018. https://arxiv.org/abs/1807.07500

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    J. A. Ellis-Monaghan and I. Moffatt. A Penrose polynomial for embedded graphs. European J. Combin. , 34(2):424--445, 2013. https://arxiv.org/abs/1106.5279, https://doi.org/10.1016/j.ejc.2012.06.009 doi:10.1016/j.ejc.2012.06.009

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.ejc.2012.06.009 2013

  21. [21]

    Gagnon-Ririe and M

    L. Gagnon-Ririe and M. B. Young. Frobenius--Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory. J. Geom. Phys. , 203:105260, 2024. https://arxiv.org/abs/2310.03566, https://doi.org/10.1016/j.geomphys.2024.105260 doi:10.1016/j.geomphys.2024.105260

    work page doi:10.1016/j.geomphys.2024.105260 2024

  22. [22]

    J.A. Green. On the structure of semigroups. Ann. of Math. (2) , 54:163--172, 1951. https://doi.org/10.2307/1969317 doi:10.2307/1969317

    work page doi:10.2307/1969317 1951

  23. [23]

    A categorification for the chromatic polynomial

    L. Helme-Guizon and Y. Rong. A categorification for the chromatic polynomial. Algebr. Geom. Topol. , 5:1365--1388, 2005. https://arxiv.org/abs/math/0412264, https://doi.org/10.2140/agt.2005.5.1365 doi:10.2140/agt.2005.5.1365

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/agt.2005.5.1365 2005

  24. [24]

    An invariant of link cobordisms from Khovanov homology

    M. Jacobsson. An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. , 4:1211--1251, 2004. https://arxiv.org/abs/math/0206303, https://doi.org/10.2140/agt.2004.4.1211 doi:10.2140/agt.2004.4.1211

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/agt.2004.4.1211 2004

  25. [25]

    F. Jaeger. On transition polynomials of 4-regular graphs. In Cycles and rays , volume 301 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , pages 123--150. Kluwer Acad. Publ., Dordrecht, 1990. https://doi.org/10.1007/978-94-009-0517-7_12 doi:10.1007/978-94-009-0517-7_12

    work page doi:10.1007/978-94-009-0517-7_12 1990

  26. [26]

    E. F. Jasso-Hernandez and Y. Rong. A categorification for the Tutte polynomial. Algebr. Geom. Topol. , 6:2031--2049, 2006. https://arxiv.org/abs/math/0512613, https://doi.org/10.2140/agt.2006.6.2031 doi:10.2140/agt.2006.6.2031

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/agt.2006.6.2031 2031

  27. [27]

    L. H. Kauffman. Categorification of chromatic, dichromatic and Penrose polynomials. Preprint, 2025. https://arxiv.org/abs/2512.21027, https://doi.org/10.48550/arXiv.2512.21027 doi:10.48550/arXiv.2512.21027

    work page doi:10.48550/arxiv.2512.21027 2025

  28. [28]

    L. H. Kauffman. State models and the Jones polynomial. Topology , 26(3):395--407, 1987. https://doi.org/10.1016/0040-9383(87)90009-7 doi:10.1016/0040-9383(87)90009-7

    work page doi:10.1016/0040-9383(87)90009-7 1987

  29. [29]

    Kelom \"a ki, A

    T. Kelom \"a ki, A. Lacabanne, D. Tubbenhauer, P. Vaz, and V.L. Zhang. On detection probabilities of link invariants. 2025. https://arxiv.org/abs/2509.05574, https://doi.org/10.48550/arXiv.2509.05574 doi:10.48550/arXiv.2509.05574

    work page doi:10.48550/arxiv.2509.05574 2025

  30. [30]

    A categorification of the Jones polynomial

    M. Khovanov. A categorification of the Jones polynomial. Duke Math. J. , 101(3):359--426, 2000. https://arxiv.org/abs/math/9908171, https://doi.org/10.1215/S0012-7094-00-10131-7 doi:10.1215/S0012-7094-00-10131-7

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1215/s0012-7094-00-10131-7 2000

  31. [31]

    Foam evaluation and Kronheimer--Mrowka theories

    M. Khovanov and L.-H. Robert. Foam evaluation and Kronheimer--Mrowka theories. Adv. Math. , 376:Paper No. 107433, 59, 2021. https://arxiv.org/abs/1808.09662, https://doi.org/10.1016/j.aim.2020.107433 doi:10.1016/j.aim.2020.107433

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2020.107433 2021

  32. [32]

    Khovanov, M

    M. Khovanov, M. Sitaraman, and D. Tubbenhauer. Monoidal categories, representation gap and cryptography. Trans. Amer. Math. Soc. Ser. B , 11:329--395, 2024. URL: https://arxiv.org/abs/2201.01805, https://doi.org/10.1090/btran/151 doi:10.1090/btran/151

    work page doi:10.1090/btran/151 2024

  33. [33]

    J. Kock. Frobenius algebras and 2 D topological quantum field theories , volume 59 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511615443 doi:10.1017/CBO9780511615443

    work page doi:10.1017/cbo9780511615443 2004

  34. [34]

    P. B. Kronheimer and T. S. Mrowka. Tait colorings, and an instanton homology for webs and foams. J. Eur. Math. Soc. (JEMS) , 21(1):55--119, 2019. https://arxiv.org/abs/1508.07205, https://doi.org/10.4171/JEMS/831 doi:10.4171/JEMS/831

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4171/jems/831 2019

  35. [35]

    S. K. Lando and A. K. Zvonkin. Graphs on surfaces and their applications . Encyclopaedia of Mathematical Sciences, volume 141. Springer, Berlin, 2004. https://doi.org/10.1007/978-3-540-38361-1 doi:10.1007/978-3-540-38361-1

    work page doi:10.1007/978-3-540-38361-1 2004

  36. [36]

    The chromatic polynomial of fatgraphs and its categorification

    M. Loebl and I. Moffatt. The chromatic polynomial of fatgraphs and its categorification. Adv. Math. , 217(4):1558--1587, 2008. https://arxiv.org/abs/math/0511557, https://doi.org/10.1016/j.aim.2007.11.016 doi:10.1016/j.aim.2007.11.016

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2007.11.016 2008

  37. [37]

    Luse and Y

    K. Luse and Y. Rong. A categorification for the Penrose polynomial. J. Knot Theory Ramifications , 20(1):141--157, 2011. https://doi.org/10.1142/S021821651100867X doi:10.1142/S021821651100867X

    work page doi:10.1142/s021821651100867x 2011

  38. [38]

    V. O. Manturov. Khovanov homology for virtual knots with arbitrary coefficients. J. Knot Theory Ramifications , 16(3):345--377, 2007. https://arxiv.org/abs/math/0601152, https://doi.org/10.1142/S0218216507005336 doi:10.1142/S0218216507005336

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218216507005336 2007

  39. [39]

    Categories generated by a trivalent vertex

    S. Morrison, E. Peters, and N. Snyder. Categories generated by a trivalent vertex. Selecta Math. (N.S.) , 23(2):817--868, 2017. URL: https://arxiv.org/abs/1501.06869, https://doi.org/10.1007/s00029-016-0240-3 doi:10.1007/s00029-016-0240-3

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00029-016-0240-3 2017

  40. [40]

    R. Penrose. Applications of negative dimensional tensors. In D. J. A. Welsh, editor, Combinatorial Mathematics and its Applications , pages 221--244. Academic Press, New York, 1971

    1971

  41. [41]

    K. Tagami. A K hovanov type invariant derived from an unoriented HQFT for links in thickened surfaces. Internat. J. Math. , 24(10):1350078, 28, 2013. URL: https://arxiv.org/abs/1311.1909, https://doi.org/10.1142/S0129167X1350078X doi:10.1142/S0129167X1350078X

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0129167x1350078x 2013

  42. [42]

    Temperley and E.H

    H.N.V. Temperley and E.H. Lieb. Relations between the ``percolation'' and ``colouring'' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ``percolation'' problem. Proc. Roy. Soc. London Ser. A , 322(1549):251--280, 1971. https://doi.org/10.1098/rspa.1971.0067 doi:10.1098/rspa.1971.0067

    work page doi:10.1098/rspa.1971.0067 1971

  43. [43]

    On rank one 2-representations of web categories

    D. Tubbenhauer. On rank one 2-representations of web categories. Algebr. Comb. 7 (2024), no. 6, 1813--1843 https://arxiv.org/abs/2307.00785, https://doi.org/10.5802/alco.389 doi:10.5802/alco.389

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.5802/alco.389 2024

  44. [44]

    Tubbenhauer

    D. Tubbenhauer. Quantum topology without topology. 2025. https://arxiv.org/abs/2506.18918

    work page arXiv 2025

  45. [45]

    Sandwich cellularity and a version of cell theory

    D. Tubbenhauer. Sandwich cellularity and a version of cell theory. Rocky Mountain J. Math. , 54(6):1733--1773, 2024. https://arxiv.org/abs/2206.06678, https://doi.org/10.1216/rmj.2024.54.1733 doi:10.1216/rmj.2024.54.1733

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1216/rmj.2024.54.1733 2024

  46. [46]

    Virtual Khovanov homology using cobordisms

    D. Tubbenhauer. Virtual Khovanov homology using cobordisms. J. Knot Theory Ramifications , 23(9):1450046, 91, 2014. https://arxiv.org/abs/1111.0609, https://doi.org/10.1142/S0218216514500461 doi:10.1142/S0218216514500461

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218216514500461 2014

  47. [47]

    Tubbenhauer and V

    D. Tubbenhauer and V. Zhang. Big data comparison of quantum invariants. 2025. To appear in Journal of Experimental Mathematics . https://arxiv.org/abs/2503.15810

    work page arXiv 2025

  48. [48]

    Unoriented topological quantum field theory and link homology

    V. Turaev and P. Turner. Unoriented topological quantum field theory and link homology. Algebr. Geom. Topol. , 6:1069--1093, 2006. https://arxiv.org/abs/math/0506229, https://doi.org/10.2140/agt.2006.6.1069 doi:10.2140/agt.2006.6.1069

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/agt.2006.6.1069 2006

  49. [49]

    Turaev and A

    V.G. Turaev and A. Virelizier. Monoidal categories and topological field theory , volume 322 of Progress in Mathematics . Birkh \"a user/Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-49834-8 doi:10.1007/978-3-319-49834-8

    work page doi:10.1007/978-3-319-49834-8 2017

  50. [50]

    S. Yamada. An invariant of spatial graphs. J. Graph Theory , 13(5):537--551, 1989. https://doi.org/10.1002/jgt.3190130503 doi:10.1002/jgt.3190130503

    work page doi:10.1002/jgt.3190130503 1989