Pith. sign in

REVIEW 3 major objections 6 minor 112 references

Quantum invariants of 3-manifolds and links: a review

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read This review assembles the evidence that the q-series invariants \hat Z_b and F_L are a coherent family: they are quantum modular, built from infinite-dimensional Verma modules, expressible as quiver series, and linked at roots of unity to W

desk verdict A useful but under-polished survey of \hat{Z} and F_L; the R-matrix formula as printed has a typo that breaks the Yang-Baxter check, so treat the equations as notes, not definitions. read the letter →

arxiv 2509.02939 v1 pith:YRHM32GQ submitted 2025-09-03 math-ph hep-thmath.GTmath.MPmath.QA

John Chae This is my paper
classification math-phhep-thmath.GTmath.MPmath.QA
keywords q-seriesinvariants3-manifoldknotquantummodularityVermamodulesknot-quivercorrespondenceADOpolynomialsLiesuperalgebras
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 review's subject is a pair of q-series invariants: $\hat{Z}_b$ for closed 3-manifolds and $F_L$ for link complements, both originating in a three-dimensional supersymmetric quantum field theory of Chern-Simons type. The paper assembles definitions, computational techniques, theorems, and conjectures to show these invariants are not isolated objects. They exhibit quantum modularity under the modular group, admit an R-matrix construction using infinite-dimensional Verma modules, expand into quiver generating series, and reduce at roots of unity to classical invariants such as the WRT invariant, ADO polynomials, and Witt invariants. A separate thread extends the whole picture to the Lie superalgebra $sl(2|1)$, where the invariants carry two Spinc labels. The goal is to present these q-series as a unified bridge between physics, low-dimensional topology, and quantum algebra.

What carries the argument

The central objects are two q-series invariants. For a plumbed 3-manifold $Y$, $\hat{Z}_b$ is a principal-value contour integral of a theta function built from the plumbing matrix; this is the object whose modular properties and root-of-unity limits are analyzed. For a link $L$, $F_L$ is defined through an inverted state sum: a braid representative is evaluated with large-color $R$-matrices acting on infinite-dimensional highest and lowest weight Verma modules of $U_q(sl(2))$ (with multicolor generalizations), then closed by a reduced quantum trace. These $R$-matrices supply the infinite-dimensional representation theory behind $F_L$, while a quiver generating series gives an alternative packaging of the sam

What would settle it

Compute $F_{4_1}(x,q)$ at $q=\zeta_5$ and compare with $(x^{1/2}-x^{-1/2}) ADO_5(4_1;x)/\Delta_{4_1}(x^5)$; any discrepancy disproves the conjectured root-of-unity connection to ADO polynomials (Conjecture 3.31).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery being reviewed is that the q-series invariants $\hat{Z}_b(Y;q)$ and $F_L(x_i,q)$ form a coherent family with a characteristic set of properties. $\hat{Z}_b$, originally predicted as the BPS partition function of a 3d $N=2$ theory on a manifold $Y$, is a convergent integral q-series conjecturally equal to the graded Euler characteristic of a homology categorifying the WRT invariant. $F_L$, defined for link complements first through plumbing and then through large-color R-matrices, obeys the same patterns of modularity, recursion, and root-of-unity specialization. The review records evidence that these series are quantum modular forms, that their perturbative

Load-bearing premise

The review's unifying picture rests on several unproved conjectures (WRT decomposition, surgery formulas, inverted Habiro series, super decomposition), and its account is only as reliable as its transcriptions: Theorem 2.5 attributes a proof to a 'Conjecture 1.1' that is never defined, so that particular attribution cannot be checked.

Editorial extensions

If this is right

  • If the WRT decomposition conjecture holds, the WRT invariant of every rational homology 3-sphere becomes a finite linear combination of radial limits of \hat Z_b, making the q-series the fundamental building block of the quantum invariant.
  • If the regularized surgery formulas hold, \hat Z_b can in principle be computed for any 3-manifold obtained by Dehn surgery on a link in S^3, going well beyond plumbed examples.
  • The theorem that F_L's \hbar-expansion agrees with the Melvin\u2013Morton\u2013Rozansky expansion implies F_L encodes the Alexander\u2013Conway function and higher perturbative data of links.
  • The quantum modularity results place each \hat Z_b into a representation of a covering of SL(2,Z), so modular transformations of false and mock theta functions transfer computations between a manifold and its orientation reversal.
  • The super extension implies that non-semisimple invariants of plumbed manifolds decompose into \hat Z_{b,c}^{sl(2|1)}, so supergroup invariants inherit the same surgery and modularity framework.

Reading between the lines

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

  • One testable extension suggested by the review: the inverted-state-sum machinery for homogeneous links may extend to arbitrary braid closures if the crossing-sign assignments are made coordinate-free, which would make F_L an invariant of all links rather than only homogeneous ones.
  • The pair of Spinc labels in \hat Z_{b,c}^{sl(2|1)} may admit an interpretation as a super analogue of Heegaard Floer correction terms; the paper does not pursue this, but the structure of its examples invites the comparison.
  • The orientation-reversal pairs of false and mock theta functions suggest the super series should also come in Weyl-symmetric pairs under y \leftrightarrow y^{-1}, z \leftrightarrow z^{-1}; this symmetry appears in the computed examples and could be promoted to a general conjecture.
  • If the quiver forms of F_K are canonical, quiver mutation could relate different surgery presentations of the same 3-manifold, giving a combinatorial check of the surgery formulas.
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

3 major / 6 minor

Summary. This paper is a review of two families of q-series invariants in 3-manifold topology: the 3-manifold invariant \hat{Z} and the link-complement invariant F_L, together with their supergroup analogues. After recapping the plumbed-manifold definition of \hat{Z}, the paper reviews quantum modularity, line operators, effective central charge, relations to Rokhlin/Witt invariants, and orientation reversal. For F_L, it reviews the large-color R-matrix construction, inverted state sums, inverted Habiro series, Dehn surgery formulas, ADO polynomials, and the knot-quiver correspondence. A final section covers the sl(2|1) generalization \hat{Z}_{b,c} and super F_K. The paper is purely a survey; no new theorems are proved.

Significance. The review is potentially useful as an entry point to a rapidly growing literature. Its strengths are the breadth of topics covered, the inclusion of many explicit formulas and examples, and the clear separation of theorems and conjectures. In particular, the presentation of the R-matrix formulation, the surgery formulas, and the supergroup extension collects material that is otherwise scattered. However, because the value of a review depends on reliable transcription, the errors identified below need to be fixed before the paper can be used as a reference. No machine-checked proofs or code are supplied, but the paper's role is expository rather than computational.

major comments (3)
  1. [Section 3.2, Eq. (31)] The large-color R-matrix is written with q^{(j'+j'+1)/2} x^{-(j'+j'+1)/2} in the displayed formula, and the same repeated-j' exponent appears in the extended R-matrix (33) and (35). The exponent must be symmetric in the two strand labels; the standard U_q(sl(2)) expression uses (j'+j+1)/2. With the printed exponent, the R-matrix is not invariant under exchanging the two strands and cannot satisfy the quantum Yang-Baxter equation stated immediately after Eq. (31). Since Theorem 3.9 and the examples in Section 3.10 depend on these matrices, the definition of F_L is not reproducible as written. Please correct the exponent and verify all R-matrix formulas against [93,94].
  2. [Theorem 2.5] The theorem states that 'Conjecture 1.1 holds for negative definite plumbed 3-manifolds', but no Conjecture 1.1 is defined anywhere in the manuscript. The only plausible reading is Conjecture 2.1 (the WRT decomposition), but the mismatch makes the attribution to [82] unverifiable. The conjecture should be explicitly renumbered or redefined before publication.
  3. [Section 3.3, Theorem 3.9] The theorem defines F_L := (x^{1/2}-x^{-1/2}) Z_inv(β_L) with a single variable x and 'the parameter associated to the open strand', while Remark 3.10 states that F_L is a function of x_1,...,x_l for an l-component link L. For l > 1, the prefactor should presumably be a product over all components (or the variables should be encoded in Z_inv). As written, the definition is ambiguous and cannot reproduce the link surgery formula in Conjecture 3.20. Please align the notation with [94].
minor comments (6)
  1. [Section 2.1] The sentence 'It was shown in [ ?] that sign of e determines...' contains an unresolved citation placeholder. Please fill in the reference.
  2. [Section 2.3, first example] The text says 'We find that m = 3', but the subsequent notation σ_{18+9} and the formula 4m = lcm(8,12,36,3) = 72 imply m = 18. This inconsistency should be corrected.
  3. [Section 2.6] The parentheticals '(cf.(4))' and '(cf.(5))' after the definitions of w(Y) and def_3(Θ) should refer to Eqs. (24) and (25), respectively, where those quantities are actually defined.
  4. [Remark 4.4] The remark says 'We will see in the origin of the diverging constant in Section 5 and 6', but the manuscript has no Section 6. This cross-reference should be corrected.
  5. [Section 5.2] The text refers to 'Theorem 2.57', which does not exist; the intended reference is presumably Theorem 5.2 in the same section.
  6. [Section 4.1, Eq. (56)] The exponent 'deg(v_s)' should presumably be 'deg(v)'; the subscript s is undefined.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the review reports prior results, and its self-citations are not load-bearing; a few referencing/transcription defects are correctness issues, not circularity.

full rationale

This is an expository review, not an original derivation. The invariants \hat Z, F_K, and F_L are introduced by quoting definitions and results from the literature, and the paper does not attempt to derive them from first principles. The only steps that could raise self-citation concerns are the author's own papers [9]–[13], used for ADO formulas, Witt invariants, cable knots, and the super knot-complement series. These are citations to separate prior papers, not to this review, and the review does not use its own conclusions as premises. The central survey content is independently anchored in [48], [93], [94], [27], [14], and other external sources. The explicitly conjectural formulas (Conjectures 2.1, 2.8, 3.17–3.20, 3.11, 4.5) are labeled as conjectures and are not presented as derived predictions. No equation in the paper is equivalent by construction to its input, and no fitted parameter is relabeled as a prediction. There are genuine verifiability defects that should be corrected but are not circularity: Theorem 2.5 cites an undefined 'Conjecture 1.1'; Section 2.1 contains an unresolved '[?]' citation; Section 5.2 refers to a nonexistent 'Theorem 2.57'; and Eq. (31) prints the same index j' in both q- and x-exponents, breaking the stated Yang-Baxter symmetry. These affect reproducibility, not circularity. Score 2 reflects the presence of several self-citations in the survey, none of which is load-bearing in a circular sense.

Assumptions & free parameters 1 free parameters · 9 assumptions · 0 invented entities

The review is an exposé of other people's constructions; the ledger therefore lists the conjectures and physical assumptions that the survey treats as established. The main risk is that many entries are open problems labeled as such, but the survey sometimes lets the reader forget that, e.g., the regularized surgery formulas in Conjectures 3.18-3.19 are not proven. The only fitted number in the review is the c_eff growth parameter m in Section 2.5.

free parameters (1)
  • c_eff parameter m(s,t) = numerical estimates, e.g., m(4,7)=3.90, m(5,5)=5.01, m(11,11)=11.33
    Section 2.5 reports c_eff = 2 + 24m^2/(4st(rst +/- 1)) where m is estimated from numerical analysis of coefficient growth; this is a fitted parameter from [54], not derived.
assumptions (9)
  • domain assumption Existence of BPS homology H^{i,j}_{BPS}(Y;b) categorifying Zhat_b
    Conjectured in [52,53] and described in Section 2.2 (Physics Story); not constructed mathematically.
  • domain assumption WRT invariant decomposes into Zhat_b (Conjecture 2.1)
    Proved for negative definite plumbed manifolds [82], open in general; presented as a central organizing fact.
  • domain assumption Superconformal index factorization I_sc = sum |W_b| Zhat_b(Y) Zhat_b(-Y;1/q) (Conjecture 2.8)
    Physics conjecture from [53], verified for examples; used in Section 2.7 for orientation reversal.
  • domain assumption Validity of Dehn surgery formulas for F_K and F_L (Theorem 3.16 and Conjectures 3.17-3.20)
    Proven for plumbed knots [48]; general and regularized versions are conjectural; the review uses them as computational tools.
  • domain assumption Inverted Habiro series for F_K (Conjecture 3.11)
    Conjectural for knots with non-monic Alexander polynomial; presented as a definitional equivalence in (39).
  • domain assumption ADO relations F_K|q=zeta_p = (x^{1/2}-x^{-1/2}) ADO_p/Delta (Conjecture 3.31 and refined Conjecture 3.32)
    Verified only for small cases; central to Section 3.8.
  • domain assumption Knot-quiver correspondence generating F_K from motivic series (Section 3.9, Eq. (45))
    Proven or extended for classes of knots; general validity is open.
  • domain assumption Existence of good chambers for super Zhat (conditions (57)-(58))
    The super series (56) is only defined when a good chamber exists; nontrivial condition from [27].
  • standard math Standard plumbing, quantum group, and modular form background
    Kirby calculus, quantum invariants, mock/false theta functions are used throughout without proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum invariants of 3-manifolds and links: a review." pith.science (2026). https://pith.science/paper/YRHM32GQ

@misc{pith2026250902939,
  author       = {Pith},
  title        = {Pith review of: Quantum invariants of 3-manifolds and links: a review},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRHM32GQ}},
  note         = {Machine review of arXiv:2509.02939}
}
abstract

We review the recent developments of quantum invariants of 3-manifolds and links: $\hat{Z}$ and $F_L$. They are $q$-series invariants originated from mathematical physics. They exhibit rich features, for example, quantum modularity, infinite dimensional Verma module structures and knot-quiver correspondence. Furthermore, they have connections to other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras.

Figures

Figures reproduced from arXiv: 2509.02939 by the authors.

Figure 1
Figure 1. Kirby-Neumann moves on plumbing trees. Move 1: blow up/down (left), move 2: [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Topological invariants at the fourth and the sixth roots of unity from the limits of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. A plumbing graph ΓK of a knot K ⊂ S 3 (left) and corresponding surgery link L(ΓK). The linking between two link components is the Hopf link. This link diagram can be transformed into a knot diagram through the Kirby moves. 3. Resurgence method: This method utilizes quantum modular property of Zˆ b(Y ; q) [20] 16 . The method aims to find a dual Ψm,r(q) ∨ of a false theta functions Ψm,a(q): Ψm,a(q) ←→ Ψm,a(q) ∨ A sta… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Plumbing graphs of T(2, 2n+ 1) (left), T(3, 3n+ 1) (right) and, T(3, 3n+ 2) (bottom). The ellipsis indicates intermediate vertices with weight −2 along the legs. Total number of −2 vertices in succession on the leg is n − 1 for T(2, 2n + 1), T(3, 3n + 1) and T(3, 3n + …
Figure 5
Figure 5. Figure 5: Braid β, more precisely (1, 1)-tangle setup [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Rˇ and Rˇ−1 matices for positive and negative crossings are shown respectively. As in the colored Jones polynomials case, the R-matrix approach utilizes a braid presen￾tation β of K in [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Crossing for the inverted state sum. shown in [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: The cigars of the Taub-NUT space of 11-dimensional spacetime that are wrapped by [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

112 extracted references · 25 canonical work pages

  1. [27]

    Ferrari, P

    F. Ferrari, P. Putrov, Supergroups, q-series and 3-manifolds, Annales Henri Poincare , Volume 25, pages 2781-2837, (2024) arXiv:2009.14196

    arXiv 2024

  2. [13]

    A supergroup series for knot complements

    J. Chae, A supergroup series for knot complements, arXiv:2508.10279

    work page Pith review arXiv

  3. [82]

    Murakami, A Proof of a Conjecture of Gukov–Pei–Putrov–Vafa, Communications in Mathematical Physics, Volume 405, article number 274, (2024)

    Y. Murakami, A Proof of a Conjecture of Gukov–Pei–Putrov–Vafa, Communications in Mathematical Physics, Volume 405, article number 274, (2024). arXiv:2302.13526

    arXiv 2024

  4. [94]

    Park, Inverted state sums, inverted Habiro series, and indefinite theta functions, arXiv:2106.03942

    S. Park, Inverted state sums, inverted Habiro series, and indefinite theta functions, arXiv:2106.03942

    arXiv

  5. [1]

    $c_{\rm eff}$ from Resurgence at the Stokes Line

    G. Adams, O. Costin, G. Dunne, S. Gukov, O. Oner, cef ffrom Resurgence at the Stokes Line, arXiv:2508.10112v1

    work page Pith review arXiv

  6. [2]

    Akutsu, T

    Y. Akutsu, T. Deguchi, and T. Ohtsuki, Invariants of colored links, Journal of Knot Theory and its Ramifications 1, no. 02, 161-184, 1992

    1992

  7. [3]

    Aganagic, Homological knot invariants from mirror symmetry, Proc

    M. Aganagic, Homological knot invariants from mirror symmetry, Proc. Int. Cong. Math. 2022 arXiv:2207.14104

    arXiv 2022

  8. [4]

    Lattice cohomology and $q$-series invariants of $3$-manifolds

    R. Akhmechet, P. Johnson, V. Krushkal, Lattice cohomology and q-series invariants of 3-manifolds, arXiv:2109.14139

    work page Pith review arXiv

Show all 112 references
  1. [5]

    Akhmechet, P

    R. Akhmechet, P. Johnson, S. Park, Knot lattice homology and q-series invariants for plumbed knot complements, arXiv:2403.14461

    arXiv

  2. [6]

    Atiyah, Topological quantum field theory, Publications mathematiques de l I.H.E.S 68 (1988), p

    M. Atiyah, Topological quantum field theory, Publications mathematiques de l I.H.E.S 68 (1988), p. 175-186

    1988

  3. [7]

    J. Baez, J. Dolan, Higher dimensional algebra and topological quantum field theory, Jour- nal of Mathematical Physics 36, 6073 (1995)

    1995

  4. [8]

    Bar-Natan, S

    D. Bar-Natan, S. Garoufalidis, On the Melvin-Morton-Rozansky conjecture, Invent. math 125, 103-133, 1996

    1996

  5. [9]

    Chae, Knot Complement, ADO Invariants and their Deformations for Torus Knots, SIGMA 16 (2020), 134, arXiv:2007.13277

    J. Chae, Knot Complement, ADO Invariants and their Deformations for Torus Knots, SIGMA 16 (2020), 134, arXiv:2007.13277

    2020 arXiv

  6. [10]

    Chae, Witt invariants from q-series, Letters in Mathematical Physics Volume 113, article number 3, (2023), arXiv:2204.02794

    J. Chae, Witt invariants from q-series, Letters in Mathematical Physics Volume 113, article number 3, (2023), arXiv:2204.02794

    2023 arXiv

  7. [11]

    Chae, A Cable Knot and BPS-Series, SIGMA 19 (2023), 002, 12 pages, arXiv:2101.11708

    J. Chae, A Cable Knot and BPS-Series, SIGMA 19 (2023), 002, 12 pages, arXiv:2101.11708

    2023 arXiv

  8. [12]

    Chae, A Cable Knot and BPS-Series II, Experimental Mathematics Volume 34, 2025, arXiv:2303:083330

    J. Chae, A Cable Knot and BPS-Series II, Experimental Mathematics Volume 34, 2025, arXiv:2303:083330

    2025

  9. [14]

    Cheng, S

    M. Cheng, S. Chun, F. Ferrari, S. Gukov, S. M. Harrison, 3d modularity, J. High Energ. Phys. 10, 2019, arXiv:1809.10148

    2019 arXiv

  10. [15]

    Cheng, I.Coman, P

    M. Cheng, I.Coman, P. Kucharski, D. Passaro, G. Sgroi, 3d Modularity Revisited, arXiv:2403.14920

  11. [16]

    Cheng, S

    M. Cheng, S. Chun, B. Feigin, F. Ferrari, S. Gukov, S. M. Harrison, D. Passaro, 3- Manifolds and VOA Characters, Communications in Mathematical Physics , Volume 405, article number 44, (2024), arXiv:2201.04640

    2024 arXiv

  12. [17]

    Cheng, Francesca Ferrari, Gabriele Sgroi, Three-manifold quantum invariants and mock theta functions, Philos

    M. Cheng, Francesca Ferrari, Gabriele Sgroi, Three-manifold quantum invariants and mock theta functions, Philos. Trans. Roy. Soc. A , 378(2163):20180439, 15, 2020. arXiv:1912.07997

    2020 arXiv

  13. [18]

    Chung, BPS invariants for Seifert manifolds, J

    H-J. Chung, BPS invariants for Seifert manifolds, J. High Energ. Phys. 113, 2020, arXiv:1811.08863. 45

    2020 arXiv

  14. [19]

    Chung, BPS invariants for a Knot in Seifert manifolds, J

    H-J. Chung, BPS invariants for a Knot in Seifert manifolds, J. High Energ. Phys. 122, 2022, arXiv:2201.08351

    2022 arXiv

  15. [20]

    Costin, G

    O. Costin, G. Dunne, A. Gruen, S. Gukov, Going to the Other Side via the Resurgent Bridge, arXiv:2310.12317

    arXiv

  16. [21]

    Crane, I

    L. Crane, I. B. Frenkel, Four dimensional topological quantum field theory, Hopf cate- gories, and the canonical bases, Journal of Mathematical Physics , 35, 5136 (1994)

    1994

  17. [22]

    Casson, C

    A. Casson, C. Gordon, On slice knots in dimension three, Proceedings of Symposia in Pure Mathematics 32, 1978,

    1978

  18. [23]

    Costantino, N

    F. Costantino, N. Geer, B. Patureau-Mirand, Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories, J. Topol. 7 (2014), no. 4, 1005-1053, arXiv:1202.3553

    2014 arXiv

  19. [24]

    S. Chun, S. Gukov, S. Park, and N. Sopenko, 3d-3d correspondence for mapping tori, J. High Energ. Phys. 09, 2020, arXiv:1911.08456

    2020 arXiv

  20. [25]

    Dunfield, S

    N. Dunfield, S. Gukov, J. Rasmussen, The Superpolynomial for Knot Homologies, Exper- imental Mathematics 15, 2, 129-160, 2006, arxiv:0505662

    2006

  21. [26]

    Ekholm, P

    T. Ekholm, P. Kucharski, P. Longhi, Multi-cover skeins, quivers, and 3d N = 2 dualities, J. High Energy Phys. 02 (2020) 018, arXiv:1910.06193

    2020 arXiv

  22. [28]

    Freed, Lectures on field theory and topology, AMS Regional conference series in math- ematics, 133, 2019

    D. Freed, Lectures on field theory and topology, AMS Regional conference series in math- ematics, 133, 2019

    2019

  23. [29]

    N. Geer, J. Kujawa, B.Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, Selecta Mathematica volume 17, pages453-504 (2011), arXiv:1001.0985

    2011 arXiv

  24. [30]

    N. Geer, J. Kujawa, B.Patureau-Mirand, Ambidextrous objects and trace functions for nonsemisimple categories, Proc. Amer. Math. Soc. 141 (2013), no. 9, arXiv:1106.4477

    2013 arXiv

  25. [31]

    Geer, B.Patureau-Mirand, Multivariable link invariants arising from sl(2|1) and the Alexander polynomial, J

    N. Geer, B.Patureau-Mirand, Multivariable link invariants arising from sl(2|1) and the Alexander polynomial, J. Pure Appl. Algebra 210 (2007), no. 1, 283-298, arXiv:math/0601291

    2007 arXiv

  26. [32]

    Geer, B.Patureau-Mirand, Multivariable link invariants arising from Lie superalgebras of type I, J

    N. Geer, B.Patureau-Mirand, Multivariable link invariants arising from Lie superalgebras of type I, J. Knot Theory Ramifications 19 (2010), no. 1, 93-115, arXiv:math/0609034

    2010 arXiv

  27. [33]

    N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re- normalized link invariants, Compos. Math. 145 (2009), no. 1, 196-212, arXiv:0711.4229

    2009 arXiv

  28. [34]

    Ekholm, A

    T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park, and P. Sulkowski, ˆZ at large N: from curve counts to quantum modularity, Communications in Mathematical Physics , Volume 396, pages 143-186, (2022), arXiv:2005.13349

    2022 arXiv

  29. [35]

    Ekholm, A

    T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park, M. Stoˇ si´ c, P. Su lkowski, Branches, quivers, and ideals for knot complements, arXiv:2110.13768 46

    arXiv

  30. [36]

    Elias and Y

    B. Elias and Y. Qi, categorification of quantum sl(2) at prime roots of unity, Adv. Math., 299(2016), 863-930

    2016

  31. [37]

    H. Fuji, S. Gukov, M. Stosic and P. Sulkowski , 3d analogs of Argyres-Douglas theories and knot homologies, J. High Energ. Phys. 175, 2013

    2013

  32. [38]

    Fenn and C

    R. Fenn and C. Rourke, On Kirby’s calculus of links, Topology Volume 18, Issue 1, 1979, Pages 1-15

    1979

  33. [39]

    Freed, Remarks on fully extended 3-dimensional topological field theories, A talk from String-Math, June, 2011

    D. Freed, Remarks on fully extended 3-dimensional topological field theories, A talk from String-Math, June, 2011

    2011

  34. [40]

    N. P. Ha, Topological invariants from quantum group Uζ(sl(2|1)) at roots of unity, arXiv:1607.03728

    arXiv

  35. [41]

    Harichurn, M

    S. Harichurn, M. Jagadale, D. Noshchenko, D. Passaro, cef ffrom surgery and modularity, arXiv:2508.10087v1

    arXiv

  36. [42]

    Garoufalidis, T

    S. Garoufalidis, T. Le , The colored Jones function is q–holonomic, Geometry and Topology 9 (2005), 1253–1293, arXiv:math/0309214

    2005 arXiv

  37. [43]

    Gruen, The sl(N ) Symmetrically Large Coloured R Matrix, arXiv:2212.05222

    A. Gruen, The sl(N ) Symmetrically Large Coloured R Matrix, arXiv:2212.05222

    arXiv

  38. [44]

    Gukov, Gauge theory and knot homologies, Fortschr

    S. Gukov, Gauge theory and knot homologies, Fortschr. Phys. 55, 2007

    2007

  39. [45]

    Gukov, Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A- Polynomial, Commun

    S. Gukov, Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A- Polynomial, Commun. Math. Phys. 255, 2005, arXiv:hep-th/0306165

    2005 arXiv

  40. [46]

    Gukov, P-S Hsin, H

    S. Gukov, P-S Hsin, H. Nakajima, S. Park, D. Pei, and N. Sopenko, Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants, Journal of Geometry and Physics, Volume 168, October 2021, 104311 arXiv:2005.05347

    2021 arXiv

  41. [47]

    Gukov, L

    S. Gukov, L. Katzarkov, J. Svoboda, ˆZb for plumbed manifolds and splice diagrams, arXiv:2304.00699

    arXiv

  42. [48]

    Gukov, C

    S. Gukov, C. Manolescu, A two-variable series for knot complements, Quantum Topol. 12, 2021, 1-109, arXiv:1904.06057

    2021 arXiv

  43. [49]

    Gukov, M

    S. Gukov, M. Marino, P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615

    arXiv

  44. [50]

    Gukov, S

    S. Gukov, S. Nawata, I. Saberi, M. Stosic, P. Sulkowski, Sequencing BPS spectra, J. High Energ. Phys. Volume 2016, article number 4, (2016) arXiv:1512.07883

    2016 arXiv

  45. [51]

    Gukov, P

    S. Gukov, P. Putrov, S. Park, Cobordism invariants from BPS q-series, Annales Henri Poincare Volume 22, pages 4173-4203, (2021), arXiv:2009.11874

    2021 arXiv

  46. [52]

    Gukov, P

    S. Gukov, P. Putrov, C. Vafa, Fivebranes and 3-manifold homology, J. High Energ. Phys. 07, 71, 2017, arXiv:1602.05302

    2017 arXiv

  47. [53]

    Gukov, D

    S. Gukov, D. Pei, P. Putrov, C. Vafa, BPS spectra and 3-manifold invariants, Journal of Knot Theory and Its Ramifications Vol. 29, No. 02, 2040003 (2020), arXiv:1701.06567

    2020 arXiv

  48. [54]

    Gukov, M

    S. Gukov, M. Jagadale, cef ffor 3d N = 2 theories, arXiv:2308.05360 47

    arXiv

  49. [55]

    Gukov, A

    S. Gukov, A. Schwarz, C. Vafa, Khovanov-Rozansky Homology and Topological Strings, Letters in Math. Phys. 74, 1, 53-74, 2005

    2005

  50. [56]

    Gruen, L

    A. Gruen, L. Suarez, A large color R-matrix for sl3, arXiv:2508.15171

    arXiv

  51. [57]

    Harichurn, A

    S. Harichurn, A. Nemethi, J. Svoboda, Delta invariants of plumbed 3-manifolds, arXiv:2412.02042v1

  52. [58]

    K. Habiro. On the quantum sl2 invariants of knots and integral homology spheres, Ge- ometry & Topology Monographs , Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001), arXiv:math/0211044

    2001 arXiv

  53. [59]

    Kapustin, E

    A. Kapustin, E. Witten, Electric-magnetic duality and the geometric Langlands program, Communications in number theory and physics Volume1,Number1,1-236,2007, arXiv:hep- th/0604151

    2007

  54. [60]

    Khovanov, A categorification of the Jones polynomial, Duke Math

    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101, 3, 359-426, 2003, arXiv:math/9908171

    2003 arXiv

  55. [61]

    Khovanov, A categorification of the colored Jones polynomial, J

    M. Khovanov, A categorification of the colored Jones polynomial, J. Knot Theory Rami- fications 14 (2005), 111–130, arXiv:math/0302060

    2005 arXiv

  56. [62]

    Khovanov, Hopfological algebra and categorification at a root of unity: the first steps, J

    M. Khovanov, Hopfological algebra and categorification at a root of unity: the first steps, J. Knot Theory Ramifications 25 (2016), no. 3, 1640006, 26,

    2016

  57. [63]

    Khovanov, sl(3) link homology, Algebraic & Geometric Topology Volume 4 (2004) 1045–1081

    M. Khovanov, sl(3) link homology, Algebraic & Geometric Topology Volume 4 (2004) 1045–1081

    2004

  58. [64]

    Khovanov, A

    M. Khovanov, A. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309-347,

    2009

  59. [65]

    Khovanov, L

    M. Khovanov, L. Rozansky, Matrix factorizations and link homology 1, Fund. Math. 199, no. 1, 1-91, 2008, arXiv:0401268

    2008

  60. [66]

    Khovanov, L

    M. Khovanov, L. Rozansky, Matrix factorizations and link homology 2, Geom. Topol. 12, no. 3, 1387-1425, 2008, arXiv:0505056

    2008

  61. [67]

    Kirby, A calculus for framed links in S3, Inventiones mathematicae, Volume 45, pages 35-56, (1978)

    R. Kirby, A calculus for framed links in S3, Inventiones mathematicae, Volume 45, pages 35-56, (1978)

    1978

  62. [68]

    Kirby, P

    R. Kirby, P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C), Inventiones math. volume 105, pages473–545 (1991)

    1991

  63. [69]

    Kirby, P

    R. Kirby, P. Melvin, X. Zhang, Quantum invariants at the sixth root of unity, Communi- cations in Mathematical Physics 151, pages607–617 (1993)

    1993

  64. [70]

    Kucharski, Quivers for 3-manifolds: the correspondence, BPS states, and 3d N = 2 theories, J

    P. Kucharski, Quivers for 3-manifolds: the correspondence, BPS states, and 3d N = 2 theories, J. High Energy Phys. 09 (2020) 075, arXiv:2005 .13394

    2020

  65. [71]

    Kucharski, M

    P. Kucharski, M. Reineke, M. Stosic, P. Sulkowski, BPS states, knots and quivers, Phys. Rev. D 96(12) (2017) 121902, arXiv:1707.02991

    2017 arXiv

  66. [72]

    Kucharski, M

    P. Kucharski, M. Reineke, M. Stosic, P. Sulkowski, Knots-quivers correspondence, Adv. Theor. Math. Phys. 23(7) (2019) 1849–1902, arXiv:1707.04017. 48

    2019 arXiv

  67. [73]

    Lauda, An introduction to diagrammatic algebra and categorified quantum sl(2), arXiv:1106.2128

    A. Lauda, An introduction to diagrammatic algebra and categorified quantum sl(2), arXiv:1106.2128

    arXiv

  68. [74]

    Liles, E

    L. Liles, E. McSpirit, Infinite families of quantum modular 3-manifold invariants, arXiv:2306.14765

    arXiv

  69. [75]

    Lawrence and D

    R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. , 3(1999), no. 1, 93-107

    1999

  70. [76]

    Lickorish, A Representation of Orientable Combinatorial 3-Manifolds, Annals of Math- ematics, Vol

    W. Lickorish, A Representation of Orientable Combinatorial 3-Manifolds, Annals of Math- ematics, Vol. 76, No. 3 (Nov., 1962), pp. 531-540

    1962

  71. [77]

    Lurie, On the classification of topological field theories, Current Developments in Math- ematics, 2009: 129-280 (2009) arXiv:0905.0465

    J. Lurie, On the classification of topological field theories, Current Developments in Math- ematics, 2009: 129-280 (2009) arXiv:0905.0465

    2009 arXiv

  72. [78]

    Milnor, D

    J. Milnor, D. Husemoller, Symmetric Bilinear Forms, A Series of Modern Surveys in Mathematics (73), Springer-Verlag 1973

    1973

  73. [79]

    Melvin, H

    P. Melvin, H. Morton, The coloured Jones function, Commun. Math. Phys. 169, 501-520, 1995

    1995

  74. [80]

    Mikhaylov, E

    V. Mikhaylov, E. Witten, Branes and Supergroups, Communications in Mathematical Physics , Volume 340, pages 699-832, (2015) arXiv:1410.1175

    2015 arXiv

  75. [81]

    Moser, Elementary surgery along a torus knot, Pacific J

    L. Moser, Elementary surgery along a torus knot, Pacific J. Math. , 38(1971), 737-745

    1971

  76. [83]

    Murakami, COLORED ALEXANDER INV ARIANTS AND CONE-MANIFOLDS,Os- aka J

    J. Murakami, COLORED ALEXANDER INV ARIANTS AND CONE-MANIFOLDS,Os- aka J. Math. 45 (2008), 541–564

    2008

  77. [84]

    Moore, N

    A. Moore, N. Tarasca, Root lattices and invariant series for plumbed 3-manifolds, arXiv:2405.14972

    arXiv

  78. [85]

    Neumann, A calculus for plumbing applied to the topology of complex surface singu- larities and degenerating complex curves, Trans

    W. Neumann, A calculus for plumbing applied to the topology of complex surface singu- larities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299-344

    1981

  79. [86]

    Nemethi, L

    A. Nemethi, L. Nicolaescu. Seiberg–Witten invariants and surface singularities, Geom. Topol. 6.1 (2002), pp. 269–328

    2002

  80. [87]

    Nawata, P

    S. Nawata, P. Ramadevi, Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, Journal of Knot Theory and Its Ramifications , Vol. 22, No. 13, 1350078 (2013)

    2013

  81. [88]

    Nawata, A

    S. Nawata, A. Oblomkov, Lectures on knot homology, arXiv:1510.01795

    arXiv

  82. [89]

    Ozsvath, Z

    P. Ozsvath, Z. Szabo, Holomorphic disks and knot invariants, Adv. Math. 186, 1, 58-116, 2004, arXiv:math/0209056

    2004 arXiv

  83. [90]

    Ozsvath, Z

    P. Ozsvath, Z. Szabo, Holomorphic disks, link invariants and the multi-variable Alexander polynomial , Alg. & Geom. Topology 8 (2008) 615-692, arXiv:math/0512286v2

    2008 arXiv

  84. [91]

    Ooguri, C

    H. Ooguri, C. Vafa, Knot Invariants and Topological Strings, Nuclear Physics B Volume 577, Issue 3, 26 June 2000, Pages 419-438, arXiv:hep-th/9912123

    2000 arXiv

  85. [92]

    Park, Higher Rank ˆZ and FK, SIGMA 16 (2020), 044, 17 pages, arXiv:1909.13002 49

    S. Park, Higher Rank ˆZ and FK, SIGMA 16 (2020), 044, 17 pages, arXiv:1909.13002 49

    2020 arXiv

  86. [93]

    Park, Large color R-matrix for knot complements and strange identities, Journal of Knot Theory and Its Ramifications Vol

    S. Park, Large color R-matrix for knot complements and strange identities, Journal of Knot Theory and Its Ramifications Vol. 29, No. 14, 2050097 (2020), arXiv:2004.02087

    2020 arXiv

  87. [95]

    Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories, arXiv:1112.1000

    C. Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories, arXiv:1112.1000

    arXiv

  88. [96]

    Rasmussen, Floer homology and knot complements, arxiv:math/0306378

    J. Rasmussen, Floer homology and knot complements, arxiv:math/0306378

    arXiv

  89. [97]

    Reshetikhin, V

    N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103, no. 3, 547-597, 1991

    1991

  90. [98]

    Reshetikhin, V

    N. Reshetikhin, V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127, no. 1, 1-26, 1990

    1990

  91. [99]

    Rokhlin, New results in the theory of four-dimensional manifolds, Doklady Acad

    V. Rokhlin, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224

    1952

  92. [100]

    Rozansky Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial, Commun

    L. Rozansky Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial, Commun. Math. Phys. 183, no. 2, 291-306, 1997

    1997

  93. [101]

    Rozansky, The Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial, Adv

    L. Rozansky, The Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial, Adv. Math 134 (1998), 1–31, arxiv:q- alg/9604005

    1998

  94. [102]

    Segal, The Definition of Conformal Field Theory, Differential Geometrical Methods in Theoretical Physics NATO ASI Series 1988

    G. Segal, The Definition of Conformal Field Theory, Differential Geometrical Methods in Theoretical Physics NATO ASI Series 1988

    1988

  95. [103]

    Stroppel, Categorification: tangle invariants and TQFTs, Proc

    C. Stroppel, Categorification: tangle invariants and TQFTs, Proc. Int. Cong. Math. 2022

    2022

  96. [104]

    Vafa, Brane/anti-Brane Systems and U (N |M ) Supergroup, arXiv:hep-th/0101218

    C. Vafa, Brane/anti-Brane Systems and U (N |M ) Supergroup, arXiv:hep-th/0101218

    arXiv

  97. [105]

    Wallace, Modifications and Cobounding Manifolds, Canadian Journal of Mathematics , Volume 12 , 1960 , pp

    A. Wallace, Modifications and Cobounding Manifolds, Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 503 - 528

    1960

  98. [106]

    Witten, Quantum field theory and the Jones polynomial, Comm

    E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, no. 3, 351-399, 1989

    1989

  99. [107]

    Witten, Fivebranes and Knots, Quantum Topology 3, 1-137, 2012, arXiv:1101.3216

    E. Witten, Fivebranes and Knots, Quantum Topology 3, 1-137, 2012, arXiv:1101.3216

    2012 arXiv

  100. [108]

    Witten, Topological quantum field theory, Comm

    E. Witten, Topological quantum field theory, Comm. Math. Phys. 117(3): 353-386 (1988)

    1988

  101. [109]

    Witten, Monopoles and Four-Manifolds, Mathematical Research Letters , 1, 769-796 (1994)

    E. Witten, Monopoles and Four-Manifolds, Mathematical Research Letters , 1, 769-796 (1994)

    1994

  102. [110]

    Witten, Khovanov Homology And Gauge Theory, arXiv:1108.3103

    E. Witten, Khovanov Homology And Gauge Theory, arXiv:1108.3103

    arXiv

  103. [111]

    Witten, Two Lectures on Gauge Theory and Khovanov Homology, arXiv:1603.03854

    E. Witten, Two Lectures on Gauge Theory and Khovanov Homology, arXiv:1603.03854

    arXiv

  104. [112]

    Zagier, Quantum Modular Forms, Clay Mathematics Proceedings Volume 12, 2010

    D. Zagier, Quantum Modular Forms, Clay Mathematics Proceedings Volume 12, 2010. 50

    2010

Pith tools

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