Pith. sign in

REVIEW 3 major objections 4 minor 52 references

Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper computes the first colored unknot and Hopf link invariants in $\mathbb{RP}^3$ using the (refined) topological vertex, obtaining series with positive-integer $q$-expansion coefficients, and conjectures they are graded Poincaré…

desk verdict Explicit topological-vertex series for local CP1×CP1, plausibly but not yet demonstrably the RP3 colored unknot and Hopf link invariants; worth refereeing for the computations alone. read the letter →

arxiv 2501.12566 v1 pith:5YMU2JT5 submitted 2025-01-22 math-ph hep-thmath.GTmath.MP

John Chae This is my paper · ORCID
classification math-phhep-thmath.GTmath.MP MSC 57K1057K1457K1681T45
keywords topologicalvertexrefinedlinkinvariantsRP3PoincareserieshomologygeometrictransitionChern-Simonstheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish that the (refined) topological vertex, applied to the toric Calabi-Yau threefold obtained from the geometric transition of the cotangent bundle of $\mathbb{RP}^3$, computes colored unknot and Hopf link invariants in $\mathbb{RP}^3$ for the first time. The invariants are infinite power series in the two Kähler parameters $Q_b$ and $Q_f$, not polynomials; every coefficient's $q$-expansion has positive integers in the unrefined case and positive-integer polynomials in $t$ in the refined case. On this evidence the paper conjectures that, after a monomial specialization of the Kähler parameters, these series are graded Poincaré series of an infinite-dimensional colored $sl(N)$ link homology theory for links in $\mathbb{RP}^3$. A sympathetic reader would care because it transplants the $S^3$ program, where topological-vertex invariants were later identified with categorified link homologies, to a new three-manifold and produces explicit, checkable series where no analogous computations existed.

What carries the argument

The load-bearing object is the partition function $Z_{\alpha\gamma^t}(Q_b,Q_f,t,q)$ built from the topological vertex $C_{\lambda\mu\nu}(q)$ (or its refined version $C_{\lambda\mu\nu}(t,q)$) on the toric graph of local $\mathbb{CP}^1\times\mathbb{CP}^1$; each edge carries a Young diagram (2d partition) and each trivalent vertex contributes a skew Schur or Macdonald function amplitude. The geometric-transition dictionary of Section 2.1 attaches a link in $\mathbb{RP}^3$ to this graph: after $T^*\mathbb{RP}^3$ transitions to local $\mathbb{CP}^1\times\mathbb{CP}^1$, the conormal of the link becomes the one-brane configuration of Figure 1 or the two-brane configuration of Figure 2. The sums over internal partitions $\lambda,\beta,\nu_1,\nu_2$, together with normalization by $Z_{\varnothing\varnothing}$, convert the vertex amplitudes into the link series.

What would settle it

Compute the same fundamental-unknot invariant independently from Chern-Simons perturbation theory on $\mathbb{RP}^3$ via the matrix-model mirror of [4], and compare its coefficient of $Q_bQ_f$ with $2\sqrt{q}/(1-q)$ from equation (2); if the rational functions differ after the paper's framing conventions are taken into account, the geometric-transition dictionary has misidentified the Lagrangian brane and the series in (1)/(8) are not $\mathbb{RP}^3$ link invariants.

Watch

Extended reading notes

Core claim

The paper's central claim is that the partition function $Z_{\alpha\gamma^t}(Q_b,Q_f,t,q)$ assembled from the topological vertex on the toric graph of local $\mathbb{CP}^1\times\mathbb{CP}^1$ (Figure 1) computes colored unknot invariants in $\mathbb{RP}^3$, and that the two-brane graph of Figure 2 does the same for the Hopf link; the refined vertex (7) yields the refined analogue. For the unknot colored by the fundamental representation the regular computation gives $Z_{\square\varnothing}=\sqrt{q}/(1-q)+Q_b\sqrt{q}/(1-q)+2Q_bQ_f\sqrt{q}/(1-q)+O(Q^4)$, and the analogous formulas (3)-(6) and (9)-(12) show the same structure for higher antisymmetric colors and for the Hopf link. The paper observes that every $q$-expansion coefficient lies in $\mathbb{Z}_+[[q]]$ in the unrefined case and in $\mathbb{Z}_+[t][[q]]$ up to half-integer powers in the refined case, and it conjectures that these are graded Poincaré series of an infinite-dimensional colored $sl(N)$ link homology theory for links in $\mathbb{RP}^3$.

Load-bearing premise

The whole computation rests on the unproven geometric-transition dictionary: the cotangent bundle of $\mathbb{RP}^3$ is assumed to deform into the toric Calabi-Yau local $\mathbb{CP}^1\times\mathbb{CP}^1$, and a link's conormal Lagrangian is assumed to become exactly the brane configuration drawn in Figures 1 and 2 and encoded in equations (1) and (8); if that dictionary fails, the computed series are not invariants of the links in $\mathbb{RP}^3$.

Editorial extensions

If this is right

  • For every symmetric or skew-symmetric color considered, the pure $Q_b$ part is a polynomial whose degree equals the number of boxes of the color, while mixed $Q_bQ_f$ terms continue to all orders; hence the invariant is an infinite series, not a finite polynomial.
  • In the limit $t=q$, each refined formula (9)-(12) reduces to the corresponding unrefined formula (2)-(6), so the regular and refined computations are mutually consistent.
  • The positivity of the $q$-expansions is the paper's evidence for Conjectures 3.1, 4.1, and 4.2, which state that after monomial specializations these series are graded Poincaré series of an infinite-dimensional colored $sl(N)$ link homology theory.
  • Compared with the $S^3$ Hopf-link computation of [14], the leading and pure-$Q_b$ terms agree up to sign, but the $Q_b^2$ term differs because of a new internal Young-diagram contribution, giving a concrete place to look for $\mathbb{RP}^3$-specific effects.

Reading between the lines

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

  • A natural next test, not performed in the paper, is a direct low-order Chern-Simons perturbation computation on $\mathbb{RP}^3$; matching the $Q_bQ_f$ coefficient of the fundamental unknot would independently confirm the geometric-transition dictionary.
  • The infinite-series form suggests the open-string sector on local $\mathbb{CP}^1\times\mathbb{CP}^1$ carries infinitely many BPS states even for a single unknot color; this would be a concrete difference from the $S^3$ case, where the invariants are polynomials.
  • The same vertex setup should extend to other lens spaces $L(p,1)$, of which $\mathbb{RP}^3$ is the $p=2$ case; the series structure found here is likely the first member of a family of similar invariants.
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 / 4 minor

Summary. The paper claims to give the first computations of colored unknot and Hopf link invariants in RP^3, using both the ordinary and refined topological vertex. The method starts from the conjectural large-N geometric transition T*RP^3 -> local CP^1 x CP^1 and represents the relevant Lagrangian branes by the toric graphs in Figures 1 and 2. The resulting partition functions Z_{\alpha\gamma^t}(Q_b,Q_f,t,q) are power series in two Kähler parameters, with rational-function coefficients in t and q. The paper observes positivity of the q-expansion coefficients for many finite truncations and conjectures that these series are graded Poincaré series of an infinite-dimensional colored sl(N) link homology for links in RP^3. The computations are explicit and pass an internal consistency check: setting t=q in the refined expressions recovers the unrefined ones (equations (9)-(12)).

Significance. If the geometric-transition dictionary were established, the paper would provide genuinely new link invariants for RP^3, with a surprising series structure that suggests a new infinite-dimensional link homology theory. The manuscript is honest about its conjectural status, and the explicit expansions with the t=q consistency checks are a useful computational starting point. However, the central identification of the computed partition functions with RP^3 link invariants is not currently supported by any independent check: the paper does not verify the closed-string amplitude against known Chern-Simons results, does not specify which of the two RP^3 unknots is being computed, and does not compare with any existing RP^3 link invariant. The significance of the results is therefore conditional on the conjectural dictionary, and the paper would need to supply such checks or be reframed as a conjectural computation under that dictionary.

major comments (3)
  1. [Section 2.1, equations (1) and (8)] The load-bearing step is the identification of the toric graphs in Figures 1 and 2 with the conormal Lagrangians of knots in RP^3 after the geometric transition. This dictionary is assumed without proof or independent check. In particular, the paper does not verify that the closed-string amplitude Z_{\phi\phi} in Appendix B reproduces the known SU(N) Chern-Simons partition function on RP^3 (see [4,51]), nor does it explain how the two Kähler parameters Q_b and Q_f are to be expressed in terms of the Chern-Simons level N and the Z_2 holonomy. The paper also never states which of the two distinct unknots in RP^3 (the null-homologous one or the generator of H_1(RP^3;Z)) is being computed. Without such a dictionary, the power series in Q_b and Q_f are not identified as link invariants in RP^3; they could be correct topological-string amplitudes for some other brane configuration. I would ask the author to add a concrete check (or a clear disclaimer that no such check exists) before the computations are called invariants of links in RP^3.
  2. [Section 4.2, Conjectures 4.1 and 4.2] The positivity and Poincaré-series conjectures are supported only by finite truncations up to O(Q^4) in Q_b,Q_f and up to q^{15} in the displayed expansions. The conjectures state positivity 'for all orders in Q' and the existence of a colored sl(N) link homology theory, but no argument or evidence beyond these truncations is given. The specializations of Q_b and Q_f to monomials in q and t are also left unspecified ('powers of the monomial factors depend on N'). While conjectures are allowed, the reader cannot test them without at least a conjectural dictionary. I recommend either proving the all-orders statement for a special case (e.g., by finding a closed product formula for the coefficients) or clearly distinguishing the verified finite-order computations from the conjectural all-orders claims.
  3. [Section 5, comparison with S^3] The comparison with S^3 results is asserted verbally rather than shown. In particular, the claim that 'the leading and Q_b terms are the same as that of the Hopf link in S^3 up to their relative sign' should be demonstrated explicitly by writing the corresponding rational functions side by side after the appropriate change of variables. As written, the reader cannot verify this comparison, and the claim is not strong enough to serve as evidence for the RP^3 interpretation. Since the paper explicitly states that there is no knot-theory result to compare with in RP^3, it would be more informative to compare with the known polynomial invariants for links in RP^3 cited in the introduction ([8,9,32]), at least for small colors.
minor comments (4)
  1. [Section 4.3, equation numbering] The displayed expression for Z_{\square\Lambda^2}(\vec{Q},t,q) is labeled '(2)' instead of the expected equation number (13). Please correct the numbering.
  2. [Section 4.3, Conjecture 4.2] Conjecture 4.2 is stated for a Hopf link L, but the formula references Z_{R\emptyset}(U,Q_b,Q_f,t,q), which uses U (unknot). The notation should be changed to reflect the link being colored, e.g., Z_{R,S}(L;Q_b,Q_f,t,q).
  3. [Abstract and Introduction] The phrase 'the first computations of colored unknots and Hopf link in RP^3' should be qualified, since the interpretation as RP^3 invariants is contingent on the conjectural geometric-transition dictionary. A phrase such as 'first computations under the conjectural large-N duality' would be more accurate.
  4. [Appendix B] The normalization Z_{\phi\phi} is stated to be symmetric under t\leftrightarrow q and to reduce to the regular case when t=q, but these properties are not demonstrated. A short explanation or factorization would help the reader trust the long expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the vertex computation is a direct evaluation of standard formulas; the RP3-link interpretation rests on an explicitly conjectural geometric-transition dictionary, which is an assumption, not a fitted or self-referential input.

full rationale

The paper's derivation chain is: assume the large-N duality T^*RP^3 -> local CP^1 x CP^1 (Section 2.1, cited to [43,51]); encode the link conormal as the brane configuration in Figures 1 and 2; then evaluate the standard topological vertex formula (1) and its refinement (8). The outputs (eqs. 2-6, 9-12) are infinite series in the formal Kahler parameters Q_b and Q_f, with no free parameters tuned to match any pre-existing RP3 invariant. The t=q reductions are internal consistency checks of the vertex algebra, not re-derivations of the output from itself. The central conjectures (Conjectures 3.1, 4.1, 4.2) are explicitly labeled as conjectures, and the paper admits 'there is no knot theory result, which we can compare the above results to' (Section 4.2). That admission flags a validation gap, not circularity. There are no load-bearing self-citations (the sole author cites no prior work of his own), no uniqueness theorem imported from the author's own papers, and no renaming of a known result as a new prediction. The computation is therefore self-contained as an algebraic exercise; its physical/topological interpretation may be unproven, but that is an external assumption rather than a circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The computation introduces no fitted numbers: Q_b and Q_f are Kähler parameters of the toric Calabi-Yau, and t,q are formal refinement variables. The central burden is carried by the untested large-N-duality dictionary and the toric graph identification. The only invented object is the conjectural link homology theory, which has no independent evidence.

assumptions (4)
  • domain assumption Large N duality: SU(N) Chern-Simons theory on T^*RP^3 is equivalent to closed A-model topological strings on the local CP^1×CP^1 after geometric transition, and knot conormals map to Lagrangian branes.
    Invoked in Section 2.1 to replace the link-in-RP^3 problem with a toric Calabi-Yau computation; no proof or independent verification is given.
  • domain assumption The toric graph with two compact edges and brane insertions (Figures 1 and 2) correctly represents the unknot and Hopf link in RP^3 after the transition.
    This is the bridge between geometry and the vertex sum in equation (1); the paper does not derive the graph from a specific knot conormal.
  • standard math The unrefined topological vertex formula and gluing rules compute the relevant open/closed topological string amplitudes.
    Used throughout Section 3; the vertex has a rigorous treatment for toric Calabi-Yau threefolds in [31], but the paper does not reprove it.
  • domain assumption The refined topological vertex formula of [22] computes the refined invariants used in Section 4.
    The refined vertex is less rigorously established than the unrefined one; the paper adopts it without qualification.
invented entities (1)
  • Infinite-dimensional colored sl(N) link homology theory for links in RP^3
    purpose: To categorify the computed power-series invariants and explain their positive integer q-expansions as Poincare series.
    Proposed in Conjectures 4.1 and 4.2; no construction, no grading, and no independent handle beyond low-order positivity and analogy with [14,46].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex." pith.science (2026). https://pith.science/paper/5YMU2JT5

@misc{pith2026250112566,
  author       = {Pith},
  title        = {Pith review of: Link in $\mathbbR\mathbbP^3$ and the Topological Vertex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YMU2JT5}},
  note         = {Machine review of arXiv:2501.12566}
}
abstract

We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants.

Figures

Figures reproduced from arXiv: 2501.12566 by the authors.

Figure 1
Figure 1. A toric graph for a Lagrangian brane in the local [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. A toric graph for a pair of Lagrangian branes in the l [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 36 canonical work pages

  1. [1]

    Aganagic, Homological knot invariants from mirror sy mmetry, Proc

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

    arXiv 2022

  2. [2]

    Awata, H

    H. Awata, H. Kanno, A. Mironov, A. Morozov, A. Morozov, No ntorus link from topological vertex, PHYSICAL REVIEW D 98, 046018 (2018) arXiv:1806.01146

    arXiv 2018

  3. [3]

    Aganagic, A

    M. Aganagic, A. Klemm, M. Marino, C. Vafa, The Topologica l Vertex, Commun. Math. Phys. Volume 254, pages 425–478, (2005) arXiv:hep-th/0305132

    arXiv 2005

  4. [4]

    Aganagic, A

    M. Aganagic, A. Klemm, M. Marino, C. Vafa, Matrix model as a mirror of Chern-Simons theory, Journal of High Energy Physics Volume 2004, JHEP02(2004) arXiv:hep-th/0211098

    arXiv 2004

  5. [5]

    On knots and links in lens spaces

    A. Cattabriga, M. Manfredi, M. Mulazzani, On knots and li nks in lens spaces, Topology and its Applications 160 (2013) 430–442 arXiv:1209.6532

    work page Pith review arXiv 2013

  6. [6]

    Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$

    D. Chen, Khovanov-type homologies of null homologous li nks in RP3, arXiv:2104.04779

    work page Pith review arXiv

  7. [7]

    Bar-Natan Homology for null homologous links in \mathbb{RP}^3

    D. Chen, Bar-Natan Homology for null homologous links in RP3, arXiv:2307.11461

    work page Pith review arXiv

  8. [8]

    A polynomial invariant for links in lens spaces

    C. Cornwell, A polynomial invariant for links in lens spa ces, Journal of Knot Theory and Its Ramifications Vol. 21, No. 06, 1250060 (2012), arXiv:1002.1543

    work page Pith review arXiv 2012

Show all 52 references
  1. [9]

    Y. V. Drobotukhina, An analogue of the Jones polynomial f or links in RP 3 and a general- ization of the Kauffman-Murasugi theorem, Algebra i Analiz , 2(1990), no. 3, 171–191

    1990

  2. [10]

    Dunfield, S

    N. Dunfield, S. Gukov, and J. Rasmussen, The Superpolyno mial for Knot Homologies, Ex- perimental Mathematics 15, 2, 129-160, 2006, arXiv:0505662

    2006

  3. [11]

    Faber, R

    C. Faber, R. Pandharipande, Hodge integrals and Gromow -Witten theory, Invent. Math 139 173, math.AG./9810173. 20

  4. [12]

    Gabrovsek, The categorification of the Kauffman bracke t Skein module of RP3, Bull

    B. Gabrovsek, The categorification of the Kauffman bracke t Skein module of RP3, Bull. Aust. Math. Soc. , 88(2013), no. 3, 407–422

    2013

  5. [13]

    Gukov, Gauge theory and knot homologies, Fortschr

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

    2007

  6. [14]

    Gukov, A

    S. Gukov, A. Iqbal, C. Koz¸ caz, Cumrun Vafa , Link Homolo gies and the Refined Topological Vertex, Commun. Math. Phys. Volume 298, pages 757–785, (2010), arXiv:0705.1368

    2010 arXiv

  7. [15]

    Gukov, A

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

    2005

  8. [16]

    Gopakumar, C

    R. Gopakumar, C. Vafa, M-Theory and Topological String s–II, arXiv:hep-th/9812127

    arXiv

  9. [17]

    Gukov, J

    S. Gukov, J. Walcher, Matrix Factorizations and Kauffman Homology, arXiv:hep-th/0512298

    arXiv

  10. [18]

    Hollowood, A

    T.J. Hollowood, A. Iqbal, C. Vafa, Matrix models, geome tric engineering and elliptic genera, JHEP 03 (2008) 069

    2008

  11. [19]

    Iqbal, A.-K

    A. Iqbal, A.-K. Kashani-Poor, Instanton counting and C hern-Simons theory, Adv. Theor. Math. Phys. 7 (2004) 457

    2004

  12. [20]

    Iqbal, A.-K

    A. Iqbal, A.-K. Kashani-Poor, SU (N ) geometries and topological string amplitudes, Adv.Theor. Math. Phys. 10 (2006) 1

    2006

  13. [21]

    Iqbal, A.-K

    A. Iqbal, A.-K. Kashani-Poor, The vertex on a strip, Adv.Theor. Math. Phys. 10 (2006) 317

    2006

  14. [22]

    Iqbal, C

    A. Iqbal, C. Kozcaz, C. Vafa, The Refined Topological Ver tex, J. High Energ. Phys. Volume 2009, JHEP10(2009), arXiv:hep-th/0701156

    2009 arXiv

  15. [23]

    JOCKERS, A

    H. JOCKERS, A. KLEMM, M. SOROUSH, Torus Knots and the Top ological Vertex, Lett Math Phys (2014) 104:953–989, arXiv:1212.0321

    2014 arXiv

  16. [24]

    Konishi, Integrality of Gopakumar-Vafa Invariants of Toric Calabi-Yau Threefolds, Publ

    Y. Konishi, Integrality of Gopakumar-Vafa Invariants of Toric Calabi-Yau Threefolds, Publ. Res. Inst. Math. Sci. 42 (2006), no. 2, pp. 605–648, arXiv:math/0504188

    2006 arXiv

  17. [25]

    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:9908171

    2003

  18. [26]

    Knot Theory Ramifica- tions 14 (2005), 111–130, arXiv:math/0302060

    Khovanov M., A categorification of the colored Jones pol ynomial, J. Knot Theory Ramifica- tions 14 (2005), 111–130, arXiv:math/0302060

    2005 arXiv

  19. [27]

    Khovanov, L

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

    2008

  20. [28]

    Khovanov, L

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

    2008

  21. [29]

    Kalfagianni, X

    E. Kalfagianni, X. Lin,The HOMFLY polynomial for links in rational homology 3-spheres, Topology Vol. 38, No. 1, pp. 95—115, 1999, arXiv:q-alg/9509010

    1999 arXiv

  22. [30]

    Khovanov, R

    M. Khovanov, R. Lipshitz, Categorical lifting of the Jo nes polynomial: a survey, arXiv:2202.02473. 21

    arXiv

  23. [31]

    Li, C-C M

    J. Li, C-C M. Liu, K. Liu, J. Zhou, A Mathematical Theory o f the Topological Vertex, Geom. Topol. 13(1): 527-621 (2009), arXiv:math/0408426

    2009 arXiv

  24. [32]

    Manolescu, M

    C. Manolescu, M. Willis, A Rasmussen invariant for link s in RP3, arXiv:2301.09764

    arXiv

  25. [33]

    V. O. Manturov, Khovanov homology for virtual knots wit h arbitrary coefficients, J. Knot Theory Ramifications, 16(2007), no. 3, 345–377, arXiv:math/0601152v3

    2007 arXiv

  26. [34]

    Marino, Chern–Simons theory, matrix models and topo logical strings, Oxford University Press 2005

    M. Marino, Chern–Simons theory, matrix models and topo logical strings, Oxford University Press 2005

    2005

  27. [35]

    Manfredi, Lift in the 3-sphere of knots and links in le ns spaces, Journal of Knot Theory and Its Ramifications Vol

    E. Manfredi, Lift in the 3-sphere of knots and links in le ns spaces, Journal of Knot Theory and Its Ramifications Vol. 23, No. 5 (2014) 1450022 (21 pages), arXiv:1312.1256

    2014 arXiv

  28. [36]

    Nawata, A

    S. Nawata, A. Oblomkov, Lectures on knot homology, Letters in Math. Phys. 74, arXiv:1510.01795

    arXiv

  29. [37]

    Okounkov, N

    A. Okounkov, N. Reshetikhin, C. Vafa, Quantum Calabi-Y au and classical crystals, Progress in Mathematics, Volume 244, 2006, Pages 597–618

    2006

  30. [38]

    Ozsvath, Z

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

    2004

  31. [39]

    Ozsvath, Z

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

    2008 arXiv

  32. [40]

    Ooguri, C

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

    2000 arXiv

  33. [41]

    P. Peng, A simple proof of Gopakumar-Vafa conjecture fo r local toric Calabi-Yau man- ifolds, Communications in Mathematical Physics Volume 276, pages 551–569, (2007), arXiv:math/0410540

    2007 arXiv

  34. [42]

    Rasmussen, Floer homology and knot complements, arx iv:math/0306378

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

    arXiv

  35. [43]

    Stevan, Torus Knots in Lens Spaces & Topological Stri ngs, Annales Henri Poincar´ e Vol- ume 16, pages 1937–1967, (2015), arXiv:1308.5509

    S. Stevan, Torus Knots in Lens Spaces & Topological Stri ngs, Annales Henri Poincar´ e Vol- ume 16, pages 1937–1967, (2015), arXiv:1308.5509

    2015 arXiv

  36. [44]

    Stroppel, Categorification: tangle invariants and T QFTs, Proc

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

    2022

  37. [45]

    Taubes, Lagrangians for the Gopakumar-Vafa conject ure, Geometry & Topology Mono- graphs 8 (2006) 73–95 arXiv:math/0201219

    C. Taubes, Lagrangians for the Gopakumar-Vafa conject ure, Geometry & Topology Mono- graphs 8 (2006) 73–95 arXiv:math/0201219

    2006 arXiv

  38. [46]

    Wang, Colored sl(N ) homology and SU (N ) representations I: the trefoil and the Hopf link, arxiv:2211.08409

    J. Wang, Colored sl(N ) homology and SU (N ) representations I: the trefoil and the Hopf link, arxiv:2211.08409

    arXiv

  39. [47]

    Webster, An introduction to categorifying quantum k not invariants, Geometry and Topol- ogy Monographs 18, 2012

    B. Webster, An introduction to categorifying quantum k not invariants, Geometry and Topol- ogy Monographs 18, 2012

    2012

  40. [48]

    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. 22

    2012 arXiv

  41. [49]

    Witten, Chern-Simons gauge theory as a string theory , In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E

    E. Witten, Chern-Simons gauge theory as a string theory , In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E. (eds) The Floer Memorial Volume. Progress in Mathematics, vol

  42. [50]

    Yu, Open/Closed BPS Correspondence and Integrality , arXiv:2307.13542v2

    S. Yu, Open/Closed BPS Correspondence and Integrality , arXiv:2307.13542v2

    arXiv

  43. [51]

    J. Yu, Z. Zong, Torus knots in Lens spaces, open Gromov-W itten invariants, and topological recursion, arXiv:2306.05326. 23

    arXiv

  44. [133]

    arXiv:hep-th/9207094

    Birkhauser Basel. arXiv:hep-th/9207094

    arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.