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State sums for some super quantum link invariants

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a MOY-type state sum over $(N|M)$-colorings computes the $U_q(\mathfrak{gl}_{N|M})$ exterior-power link invariants, and that the renormalized $N|1$ invariant agrees with Kashaev invariants at $q=e^{i\pi/(N+1)}$.

desk verdict The state sum is real and worth having, but the proof of Theorem 2.11 is missing one local relation—fixable, but currently incomplete. read the letter →

arxiv 1909.02305 v1 pith:3WGYW277 submitted 2019-09-05 math.QA math.COmath.GT

Louis-Hadrien Robert , Emmanuel Wagner This is my paper · ORCID
classification math.QAmath.COmath.GT MSC 17B3757M27
keywords statesumssuperquantumgroupslinkinvariantsMOYgraphsexteriorpowersKashaevnon-semisimpleq-binomialidentities
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

The paper sets out to turn the algebraic Reshetikhin–Turaev invariants attached to exterior powers of the standard representation of the super quantum group $U_q(\mathfrak{gl}_{N|M})$ into explicit combinatorial state sums. Its main theorem states that for any MOY graph — an oriented planar trivalent graph with edge labels — the algebraic evaluation equals a sum over colorings of the graph by $N$ cyclic subgraphs and $M$ subgraphs, weighted by powers of $q$ and quantum binomials. Two corollaries follow: the invariant depends only on $N-M$, and it is symmetric under $q\leftrightarrow q^{-1}$. In the non-semisimple case $M=1$ with all strands labeled $N$, the quantum dimension vanishes, so the paper renormalizes the state sum to define $Q_{N|1}$ and proves that at $q=e^{i\pi/(N+1)}$ it matches the normalized colored Jones invariant of the mirror link, hence the Kashaev invariants. The authors state this combinatorial presentation as the natural input for future categorifications and computations.

What carries the argument

The workhorse is the $(N|M)$-coloring state sum. A coloring splits each edge label of a MOY graph into contributions from $N$ cyclic sub-MOY graphs and $M$ sub-MOY graphs; the weight of a coloring $c$ is $(-1)^{s(c)} q^{w_s(c)+w_{\rho}(c)} m(c)$, where $m(c)$ is a product of quantum binomials, $w_s(c)$ records signed intersections of the two parts at split vertices, $w_{\rho}(c)$ records rotations of cabled circles, and $s(c)$ is a parity. The proof that this state sum reproduces the algebraic invariant proceeds by checking the local MOY relations — the circle evaluation $\binom{N-M}{k}$, the merge/$\theta$ relations, and the ladder identities — using q-binomial addition formulas such as Proposition 1.8.

What would settle it

Take the q-binomial identity used in Proposition 4.7, $\genfrac{[}{]}{0pt}{}{i+j}{i}\genfrac{[}{]}{0pt}{}{N-1-(i+j)}{N-i}\genfrac{[}{]}{0pt}{}{-1}{i}\genfrac{[}{]}{0pt}{}{N}{j} = \genfrac{[}{]}{0pt}{}{i+j}{i}\genfrac{[}{]}{0pt}{}{N-1-(i+j)}{N-j}\genfrac{[}{]}{0pt}{}{-1}{j}\genfrac{[}{]}{0pt}{}{N}{i}$, and compute the difference of the two sides as a Laurent polynomial in $q$ for $(N,i,j)=(6,4,3)$; the identity is true exactly if that difference is the zero polynomial.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is Theorem 2.11: for every MOY graph $\Gamma$, the algebraic Reshetikhin–Turaev evaluation equals the combinatorial evaluation $\langle \Gamma \rangle_{N|M}$, defined as a sum over $(N|M)$-colorings $c=(\Gamma^E,\Gamma^S)$ of weight $(-1)^{s(c)} q^{w_s(c)+w_{\rho}(c)} m(c)$. The proof verifies that this sum obeys the same multiplicativity property and the same local MOY relations as the algebraic invariant; the q-binomial identities of Section 1 supply the circle, merge, and ladder evaluations. From Theorem 2.11 the authors conclude that both evaluations depend only on $N-M$ and are symmetric in $q$ and $q^{-1}$. In the non-semisimple case $M=1$, label $N$, they define a renormalized invariant $Q_{N|1}$ using a marked point on a strand and prove (Theorem 4.10) that at $q=e^{i\pi/(N+1)}$ it coincides with the normalized colored Jones invariant $J'_N$ of the mirror link, a root-of-unity quantum-binomial identity being the last step.

Load-bearing premise

The load-bearing premise is that a certain product of quantum analogues of binomial coefficients, displayed in the proof of Proposition 4.7, equals a symmetrically written second product for all $0\le i,j\le N$; the equality is asserted without derivation, and if it is false the proof that $Q_{N|1}$ is a link invariant collapses.

Editorial extensions

If this is right

  • For every labeled link $L$, the link invariant $P_{N|M}(L)$ defined from the state sum depends only on $N-M$, so invariants from different super ranks with the same difference coincide; this also identifies exterior-power invariants of $\mathfrak{gl}_{N|M}$ with symmetric-power invariants of $\mathfrak{gl}_{M|N}$ up to mirroring.
  • The $N|1$ renormalized invariant $Q_{N|1}$ is a nontrivial renormalization of a vanishing-quantum-dimension invariant, and at $q=e^{i\pi/(N+1)}$ it equals $J'_N(\overline{L})$, placing it in the Kashaev family that appears in the volume conjecture.
  • The invariants are now defined by explicit finite sums over subsets and multisubsets, so they can be evaluated combinatorially without building intertwiners.
  • The state sum factors into a positive multiplicity and a signed power of $q$, and the authors propose this presentation as a starting point for categorifying the Alexander/Kashaev family.

Reading between the lines

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

  • One consequence the paper leaves implicit is algorithmic: the color-sum formula is a finite sum over subsets and multisubsets, so $P_{N|M}$ and $Q_{N|1}$ can be evaluated by direct enumeration on a braid diagram.
  • Because the same polynomial arises from many pairs $(N,M)$ with fixed difference, the state-sum presentations are not unique; if categorified separately, they could produce distinct homological refinements of the same invariant, a possibility the paper cites as an open conjecture.
  • The root-of-unity specialization suggests a testable ladder: the same marked state sum could be evaluated at other roots of unity to produce Alexander-like invariants beyond the Kashaev case, though the paper does not pursue this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper defines a combinatorial state sum ⟨Γ⟩_{N|M} for oriented planar trivalent MOY graphs, where a coloring consists of N cyclic sub-MOY graphs and M arbitrary sub-MOY graphs, and claims in Theorem 2.11 that it agrees with the Reshetikhin–Turaev invariant ⟪Γ⟫_{N|M} attached to exterior powers of the standard representation of U_q(gl_{N|M}). The proof strategy is to verify the local relations of Proposition A.9 and invoke Wu's completeness theorem (Proposition A.10). Section 3 converts the graph evaluations into an oriented link invariant P_{N|M}, and Section 4 specializes to M=1 and uniform label N, where the relevant quantum dimension vanishes, defines a renormalized invariant Q_{N|1} for marked braids, and proves that at q=e^{iπ/(N+1)} it agrees with the normalized colored Jones polynomial J'_N of the mirror link, connecting the construction to Kashaev invariants.

Significance. If the central equality is established, the paper gives a genuinely parameter-free, explicit state-sum realization of super quantum link invariants that depends only on N-M, and it does so uniformly for semisimple and non-semisimple specializations. The gl_{1|1} case recovers the Alexander polynomial, the renormalized family Q_{N|1} is a natural generalization, and the explicit algebraic morphisms in Appendix A make the comparison with the representation theory checkable. The claimed independence of N-M is a strong structural statement with a concrete proof strategy, and the final root-of-unity comparison with Kashaev invariants is a falsifiable and interesting result. The manuscript is carefully organized, and the combinatorial calculus is sufficiently explicit that the missing checks can be supplied in a revision.

major comments (3)
  1. [Section 2, Theorem 2.11 and Proposition A.9] The verification of the hypotheses of Proposition A.10 is incomplete. Lemmas 2.14, 2.15, Corollary 2.17, Corollary 2.19, and Corollary 2.21 establish relations (11), (12), (13), (14), and (17) of Proposition A.9 respectively, but no argument in Section 2 establishes relation (16). Lemma 2.20 and Lemma 2.22 prove different two-term identities, and Corollary 2.21 is the sum relation (17); I do not see a specialization that yields the two-term form of (16). Since Proposition A.10 asserts that exactly this list of relations is sufficient to compute every MOY graph, the equality ⟪Γ⟫_{N|M} = ⟨Γ⟩_{N|M} is not established as written. Please add a direct verification of (16) for the combinatorial evaluation or an explicit derivation of (16) from the relations that are proved.
  2. [Section 4, Proposition 4.7] The proof of invariance under conjugation by σ1 asserts without derivation that the two products of quantum binomials are equal. This equality is load-bearing for the well-definedness of Q_{N|1} and hence for Theorem 4.10. The identity is in fact true; for instance, the suspected counterexample with N=6, i=4, j=3 reduces to a valid q-binomial identity after the common factor is removed. Nevertheless, the manuscript should supply a proof, for example by applying [r choose s] = (-1)^s [s-r-1 choose s] and then the standard identity [N choose j][j choose N-i] = [N choose i][i choose N-j].
  3. [Section 4, proof of Proposition 4.7] The reduction step 'One can actually suppose (see [SW17, Section 2.3.1]) that Γ is equal to a linear combination ...' is invoked very tersely. The displayed double sums for ⟨β1⟩ and ⟨β2⟩ require that the same coefficients λij appear for both closures, and the admissible ranges of i and j need to be stated precisely. Please expand this reduction so that the σ1-conjugation invariance is fully checkable.
minor comments (6)
  1. [Lemma 2.18] In the proof of Lemma 2.18, the sentence 'A coloring c = (ΓS,ΓS) of Γ' should read 'A coloring c = (ΓE,ΓS) of Γ'; as written the notation is inconsistent.
  2. [Lemma 2.14] In the computation of the circle evaluation, the degrees in the displayed sum appear to be interchanged: one should sum q^{deg_N(X)} over subsets X of /llbracketN/rrbracket and q^{deg_M(Y)} over multi-subsets Y of /llbracketM/rrbracket, matching Proposition 1.14. Please correct the notation.
  3. [Lemma 2.22] The proof begins 'either the two rungs receive the different colors or the receive different colors'; the intended dichotomy is 'the same color or different colors'.
  4. [Theorem 3.4] The proof of invariance under Reidemeister moves is delegated to [MOY98] in a single sentence. A short indication of how the identities of Section 2 imply the three Reidemeister moves would make the paper more self-contained.
  5. [Section 4, Proposition 4.7] The summation index n in the displayed expansions of ⟨β1⟩ and ⟨β2⟩ is not defined at that point; it should presumably be N, consistent with the labels N+i and N−j elsewhere in the proof.
  6. [Abstract] The phrase 'and explicit the relation with Kashaev invariants' should be rephrased, for example 'and make explicit the relation with Kashaev invariants'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the combinatorial evaluation is verified against the algebraic RT invariant through local relations, with sufficiency imported from Wu14 and not from the authors' own prior work.

full rationale

The paper's central claim is Theorem 2.11, asserting equality between the algebraic Reshetikhin–Turaev evaluation ⟪Γ⟫_{N|M} and the newly defined combinatorial state sum ⟨Γ⟩_{N|M}. The proof does not fit the combinatorial evaluation to the algebraic one; it checks that the state sum satisfies the multiplicativity property and the local relations (11)–(17), then invokes Proposition A.10 from Wu [Wu14] to conclude that these relations determine the invariant. This is a standard, externally grounded deduction, not a self-citation chain. The authors' previous papers [RW17], [RW18], [RW19] are cited for background, for similar definitions in the symmetric case, and for categorification context, but no load-bearing step is justified solely by those citations. The only citations imported as mathematical facts are to computational identities [KC02], to the MOY calculus [MOY98], and to renomalization/Kashaev results [GPM08, GPM10, QS15], all external to the authors. The paper even notes that an alternative proof of the Kashaev relation appears in [GPM08], so that result is not claimed as new by self-citation. The proof gaps noted by a careful reader—for instance the unproved two-term relation (16) in Proposition A.9 and the product-of-binomials identity in Proposition 4.7—are concerns about completeness or correctness of particular verifications, not instances where a prediction reduces by construction to an input. There are no fitted parameters, no quantities defined in terms of the targets they are supposed to predict, and no renaming of a known empirical pattern as a new derivation. Accordingly the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard quantum group representation theory, on the sufficiency of the local MOY relations cited from Wu14, and on the reduction of four-ended graphs to a normal form cited from Stosic-Wedrich. These are external published results, not entities invented for this paper. There are no fitted free parameters.

assumptions (3)
  • domain assumption The local relations of Proposition A.9 together with multiplicativity suffice to compute the evaluation of any MOY graph (Proposition A.10, cited to [Wu14]).
    This reduction underlies the proof of Theorem 2.11; the paper does not prove it, and the super setting might require additional relations.
  • domain assumption The morphisms defined in Appendix A (Λ, Y, ∪, ∩) are U_q(gl_{N|M})-module morphisms and give a Reshetikhin-Turaev functor (Propositions A.7, A.8), with proofs omitted as direct verifications.
    This establishes that the algebraic evaluation ⟪Γ⟫_{N|M} is well-defined and satisfies the local relations used in Section 2.
  • domain assumption Any MOY graph with four ends is a linear combination of graphs of the form shown in the proof of Proposition 4.7, cited to [SW17, Section 2.3.1].
    This is used to reduce the check of invariance under conjugation by σ1 to a finite family of graphs.

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Pith. "Pith review of State sums for some super quantum link invariants." pith.science (2026). https://pith.science/paper/3WGYW277

@misc{pith2026190902305,
  author       = {Pith},
  title        = {Pith review of: State sums for some super quantum link invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3WGYW277}},
  note         = {Machine review of arXiv:1909.02305}
}
abstract

We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the relation with Kashaev invariants.

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Works this paper leans on

38 extracted references · 14 canonical work pages

  1. [1]

    Invariants of colored links

    Yasuhiro Akutsu, Tetsuo Deguchi, and Tomotada Ohtsuki. Invariants of colored links. J. Knot Theory Ramifications , 1(2):161--184, 1992. http://dx.doi.org/10.1142/S0218216592000094 doi:10.1142/S0218216592000094

    doi:10.1142/s0218216592000094 1992

  2. [2]

    Remarks on coloured triply graded link invariants

    Sabin Cautis. Remarks on coloured triply graded link invariants. Algebr. Geom. Topol. , 17(6):3811--3836, 2017. http://dx.doi.org/10.2140/agt.2017.17.3811 doi:10.2140/agt.2017.17.3811

    doi:10.2140/agt.2017.17.3811 2017

  3. [3]

    Ellis, Ina Petkova, and Vera V\' e rtesi

    Alexander P. Ellis, Ina Petkova, and Vera V\' e rtesi. Quantum gl _ 1|1 and tangle F loer homology. Adv. Math. , 350:130--189, 2019. http://dx.doi.org/10.1016/j.aim.2019.04.023 doi:10.1016/j.aim.2019.04.023

    work page doi:10.1016/j.aim.2019.04.023 2019

  4. [4]

    Quadruply-graded colored homology of knots

    Eugene Gorsky, Sergei Gukov, and Marko Sto s i\' c . Quadruply-graded colored homology of knots. Fund. Math. , 243(3):209--299, 2018. http://dx.doi.org/10.4064/fm30-11-2017 doi:10.4064/fm30-11-2017

    work page doi:10.4064/fm30-11-2017 2018

  5. [5]

    On the colored HOMFLY - PT , multivariable and K ashaev link invariants

    Nathan Geer and Bertrand Patureau-Mirand. On the colored HOMFLY - PT , multivariable and K ashaev link invariants. Commun. Contemp. Math. , 10(suppl. 1):993--1011, 2008. http://dx.doi.org/10.1142/S0219199708003125 doi:10.1142/S0219199708003125

    doi:10.1142/s0219199708003125 2008

  6. [6]

    Multivariable link invariants arising from L ie superalgebras of type I

    Nathan Geer and Bertrand Patureau-Mirand. Multivariable link invariants arising from L ie superalgebras of type I . J. Knot Theory Ramifications , 19(1):93--115, 2010. http://dx.doi.org/10.1142/S0218216510007784 doi:10.1142/S0218216510007784

    doi:10.1142/s0218216510007784 2010

  7. [7]

    Modified quantum dimensions and re-normalized link invariants

    Nathan Geer, Bertrand Patureau-Mirand, and Vladimir Turaev. Modified quantum dimensions and re-normalized link invariants. Compos. Math. , 145(1):196--212, 2009. http://dx.doi.org/10.1112/S0010437X08003795 doi:10.1112/S0010437X08003795

    doi:10.1112/s0010437x08003795 2009

  8. [8]

    R. M. Kashaev. A link invariant from quantum dilogarithm. Modern Phys. Lett. A , 10(19):1409--1418, 1995. http://dx.doi.org/10.1142/S0217732395001526 doi:10.1142/S0217732395001526

    doi:10.1142/s0217732395001526 1995

Show all 38 references
  1. [9]

    Kauffman

    Louis H. Kauffman. Formal knot theory , volume 30 of Mathematical Notes . Princeton University Press, Princeton, NJ, 1983

    1983

  2. [10]

    Kauffman

    Louis H. Kauffman. State models and the J ones polynomial. Topology , 26(3):395--407, 1987. http://dx.doi.org/10.1016/0040-9383(87)90009-7 doi:10.1016/0040-9383(87)90009-7

    1987 doi

  3. [11]

    Quantum calculus

    Victor Kac and Pokman Cheung. Quantum calculus . Universitext. Springer-Verlag, New York, 2002. http://dx.doi.org/10.1007/978-1-4613-0071-7 doi:10.1007/978-1-4613-0071-7

    2002 doi

  4. [12]

    A categorification of the J ones polynomial

    Mikhail Khovanov. A categorification of the J ones polynomial. Duke Math. J. , 101(3):359--426, 2000. http://dx.doi.org/10.1215/S0012-7094-00-10131-7 doi:10.1215/S0012-7094-00-10131-7

    2000 doi

  5. [13]

    Matrix factorizations and link homology

    Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. Fund. Math. , 199(1):1--91, 2008. http://dx.doi.org/10.4064/fm199-1-1 doi:10.4064/fm199-1-1

    2008 doi

  6. [14]

    Foam evaluation and K ronheimer-- M rowka theories, 2018

    Mikhail Khovanov and Louis-Hadrien Robert. Foam evaluation and K ronheimer-- M rowka theories, 2018. http://arxiv.org/abs/1808.09662 arXiv:1808.09662

    2018 arXiv

  7. [15]

    Spiders for rank 2 L ie algebras

    Greg Kuperberg. Spiders for rank 2 L ie algebras. Comm. Math. Phys. , 180(1):109--151, 1996. URL: http://projecteuclid.org/getRecord?id=euclid.cmp/1104287237

    1996

  8. [16]

    The colored J ones polynomials and the simplicial volume of a knot

    Hitoshi Murakami and Jun Murakami. The colored J ones polynomials and the simplicial volume of a knot. Acta Math. , 186(1):85--104, 2001. http://dx.doi.org/10.1007/BF02392716 doi:10.1007/BF02392716

    2001 doi

  9. [17]

    On combinatorial link F loer homology

    Ciprian Manolescu, Peter Ozsv\' a th, Zolt\' a n Szab\' o , and Dylan Thurston. On combinatorial link F loer homology. Geom. Topol. , 11:2339--2412, 2007. http://dx.doi.org/10.2140/gt.2007.11.2339 doi:10.2140/gt.2007.11.2339

    2007 doi

  10. [18]

    Homfly polynomial via an invariant of colored plane graphs

    Hitoshi Murakami, Tomotada Ohtsuki, and Shuji Yamada. Homfly polynomial via an invariant of colored plane graphs. Enseign. Math. (2) , 44(3-4):325--360, 1998

    1998

  11. [19]

    A combinatorial approach to functorial quantum sl _k knot invariants

    Volodymyr Mazorchuk and Catharina Stroppel . A combinatorial approach to functorial quantum sl _k knot invariants. Am. J. Math. , 131(6):1679--1713, 2009. http://dx.doi.org/10.1353/ajm.0.0082 doi:10.1353/ajm.0.0082

    2009 doi

  12. [20]

    sl (N) -link homology (N 4) using foams and the K apustin-- L i formula

    Marco Mackaay, Marko Sto s i \'c , and Pedro Vaz. sl (N) -link homology (N 4) using foams and the K apustin-- L i formula. Geom. Topol. , 13(2):1075--1128, 2009. http://dx.doi.org/10.2140/gt.2009.13.1075 doi:10.2140/gt.2009.13.1075

    2009 doi

  13. [21]

    Holomorphic disks and knot invariants

    Peter Ozsv\' a th and Zolt\' a n Szab\' o . Holomorphic disks and knot invariants. Adv. Math. , 186(1):58--116, 2004. http://dx.doi.org/10.1016/j.aim.2003.05.001 doi:10.1016/j.aim.2003.05.001

    2004 doi

  14. [22]

    A cube of resolutions for knot F loer homology

    Peter Ozsv\' a th and Zolt\' a n Szab\' o . A cube of resolutions for knot F loer homology. J. Topol. , 2(4):865--910, 2009. http://dx.doi.org/10.1112/jtopol/jtp032 doi:10.1112/jtopol/jtp032

    2009 doi

  15. [23]

    Hoel Queffelec, David E. V. Rose, and Antonio Sartori. Annular evaluation and link homology, 2018. http://arxiv.org/abs/1802.04131 arXiv:1802.04131

    2018 arXiv

  16. [24]

    A note on the gl (m | n) link invariants and the HOMFLY-PT polynomial, 2015

    Hoel Queffelec and Antonio Sartori . A note on the gl (m | n) link invariants and the HOMFLY-PT polynomial, 2015. http://arxiv.org/abs/1506.03329 arXiv:1506.03329

    2015 arXiv

  17. [25]

    Rasmussen

    Jacob A. Rasmussen. Floer homology and knot complements . ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)--Harvard University. URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3091665

    2003

  18. [26]

    A closed formula for the evaluation of sl _n -foams

    Louis-Hadrien Robert and Emmanuel Wagner. A closed formula for the evaluation of sl _n -foams. To appear in Quantum Topology, 2017. http://arxiv.org/abs/1702.04140 arXiv:1702.04140

    2017 arXiv

  19. [27]

    Symmetric K hovanov-- R ozansky link homologies, 2018

    Louis-Hadrien Robert and Emmanuel Wagner. Symmetric K hovanov-- R ozansky link homologies, 2018. http://arxiv.org/abs/1801.02244 arXiv:1801.02244

    2018 arXiv

  20. [28]

    A quantum categorification of the A lexander polynomial, 2019

    Louis-Hadrien Robert and Emmanuel Wagner. A quantum categorification of the A lexander polynomial, 2019. http://arxiv.org/abs/1902.05648 arXiv:1902.05648

    2019 arXiv

  21. [29]

    The A lexander polynomial as quantum invariant of links

    Antonio Sartori. The A lexander polynomial as quantum invariant of links. Ark. Mat. , 53(1):177--202, 2015. http://dx.doi.org/10.1007/s11512-014-0196-5 doi:10.1007/s11512-014-0196-5

    2015 doi

  22. [30]

    Category O and sl(k) link invariants

    Joshua Sussan. Category O and sl(k) link invariants . ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)--Yale University. URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3267369

    2007

  23. [31]

    Rational links and DT invariants of quivers , 2017

    Marko Sto s i \'c and Paul Wedrich . Rational links and DT invariants of quivers , 2017. http://arxiv.org/abs/1711.03333 arXiv:1711.03333

    2017 arXiv

  24. [32]

    Vladimir G. Turaev. State sum models in low-dimensional topology. In Proceedings of the I nternational C ongress of M athematicians, V ol. I , II ( K yoto, 1990) , pages 689--698. Math. Soc. Japan, Tokyo, 1991

    1990

  25. [33]

    V. G. Turaev. Quantum invariants of knots and 3-manifolds , volume 18 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, 1994

    1994

  26. [34]

    Super q - H owe duality and web categories

    Daniel Tubbenhauer, Pedro Vaz, and Paul Wedrich. Super q - H owe duality and web categories. Algebr. Geom. Topol. , 17(6):3703--3749, 2017. http://dx.doi.org/10.2140/agt.2017.17.3703 doi:10.2140/agt.2017.17.3703

    2017 doi

  27. [35]

    Oleg Ya. Viro. Quantum relatives of the A lexander polynomial. Algebra i Analiz , 18(3):63--157, 2006. http://dx.doi.org/10.1090/S1061-0022-07-00956-9 doi:10.1090/S1061-0022-07-00956-9

    2006 doi

  28. [36]

    Quantum field theory and the J ones polynomial

    Edward Witten. Quantum field theory and the J ones polynomial. Comm. Math. Phys. , 121(3):351--399, 1989

    1989

  29. [37]

    A colored sl (N) homology for links in S^ 3

    Hao Wu. A colored sl (N) homology for links in S^ 3 . Diss. Math. , 499:215 p., 2014. http://dx.doi.org/10.4064/dm499-0-1 doi:10.4064/dm499-0-1

    2014 doi

  30. [38]

    Quantum ( sl _ n , v_ n ) link invariant and matrix factorizations

    Yasuyoshi Yonezawa. Quantum ( sl _ n , v_ n ) link invariant and matrix factorizations. Nagoya Math. J. , 204:69--123, 12 2011. http://dx.doi.org/10.1215/00277630-1431840 doi:10.1215/00277630-1431840

    2011 doi

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