Matrix model and dimensions at hypercube vertices

A. Yu. Morozov   ; A. A. Morozov   ; A. V. Popolitov

Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 1, pp. 115-163

Résumé

We consider correlation functions in the Chern–Simons theory (knot polynomials) using an approach in which each knot diagram is associated with a hypercube. The number of cycles into which the link diagram is decomposed under different resolutions plays a central role. Certain functions of these numbers are further interpreted as dimensions of graded spaces associated with hypercube vertices, but finding these functions is a somewhat nontrivial problem. It was previously suggested to solve this problem using the matrix model technique by analogy with topological recursion. We develop this idea and provide a wide collection of nontrivial examples related to both ordinary and virtual knots and links. The most powerful version of the formalism freely connects ordinary knots/links with virtual ones. Moreover, it allows going beyond the limits of the knot-related set of $(2,2)$-valent graphs.

Détail

Keywords: Chern–Simons theory, knot theory, virtual knot, matrix model.

Bibliographie

[1] V. F. R. Jones, “A polynomial invariant for knots via von Neumann algebras”, Bull. Amer. Math. Soc. (N. S.), 12:1 (1985), 103–111 ; “On knot invariants related to some statistical mechanical models”, Pacific J. Math., 137:2 (1989), 311–334 ; L. H. Kauffman, Knots and Physics, World Sci., Singapore, 1991 ; P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, “A new polynomial invariant of knots and links”, Bull. Amer. Math. Soc. (N. S.), 12:2 (1985), 239–246 ; J. H. Przytycki, K. P. Traczyk, “Invariants of links of Conway type”, Kobe J. Math., 4:2 (1987), 115–139 | DOI | MR | Zbl | DOI | MR | Zbl | MR | DOI | MR | Zbl | MR | Zbl

[2] S.-S. Chern, J. Simons, “Some cohomology classes in principal fiber bundles and their application to Riemannian geometry”, Proc. Nat. Acad. Sci. U.S.A., 68:4 (1971), 791–794 ; A. S. Schwarz, “New topological invariants arising in the theory of quantized fields”, Topology and Its Applications, Talk at the International Topological Conference. Abstracts. Part 2 (Baku, Azerbaijan, October 3–8, 1987), ed. S. P. Novikov, AMS, Providence, RI, 1993 ; E. Witten, “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121:3 (1989), 351–399 ; M. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press, Cambridge, 1990 | DOI | MR | Zbl | Zbl | DOI | MR | Zbl | DOI | MR

[3] L. H. Kauffman, “Virtual knot theory”, European J. Combin., 20:7 (1999), 663–690, arXiv: ; R. Fenn, D. P. Ilyutko, L. H. Kauffman, V. O. Manturov, “Unsolved problems in virtual knot theory and combinatorial knot theory”, Knots in Poland III. Part III (Stefan Banach International Mathematical Center, Warsaw, Poland, July 18–25, 2010 and Bȩdlewo, Poland, July 25 – August 4, 2010), Banach Center Publications, 103, eds. J. H. Przytycki, P. Traczyk, Polish Acad. Sci. Inst. Math., Warszawa, 2014, 9–61, arXiv: math/98110281409.2823 | DOI | MR | Zbl | DOI | MR | Zbl

[4] M. Khovanov, “A categorification of the Jones polynomial”, Duke Math. J., 101:3 (2000), 359–426, arXiv: math/9908171 | DOI | MR | Zbl

[5] M. Khovanov, L. Rozansky, “Matrix factorizations and link homology”, Fund. Math., 199:1 (2008), 1–91, arXiv: ; “Matrix factorizations and link homology. II”, Geom. Topol., 12:3 (2008), 1387–1425, arXiv: ; “Virtual crossings, convolutions and a categorification of the $\mathrm{SO}(2N)$ Kauffman polynomial”, J. Gökova Geom. Topol., 1 (2007), 116–214, arXiv: ; N. Carqueville, D. Murfet, “Computing Khovanov–Rozansky homology and defect fusion”, Algebr. Geom. Topol., 14:1 (2014), 489–537, arXiv: math/0401268math/0505056math/07013331108.1081 | DOI | MR | Zbl | DOI | MR | Zbl | MR | DOI | MR | Zbl

[6] S. Gukov, A. Schwarz, C. Vafa, “Khovanov–Rozansky homology and topological strings”, Lett. Math. Phys., 74:1 (2005), 53–74, arXiv: ; N. M. Dunfield, S. Gukov, J. Rasmussen, “The superpolynomial for knot homologies”, Exp. Math., 15:2 (2006), 129–159, arXiv: ; M. Aganagic, Sh. Shakirov, Knot homology from refined Chern–Simons theory, arXiv: ; “Refined Chern–Simons theory and knot homology”, String-Math 2011, Proceedings of Symposia in Pure Mathematics, 85, eds. J. Block, J. Distler, R. Donagi, E. Sharpe, AMS, Providence, RI, 2012, 3–31, arXiv: ; P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, A. Smirnov, “Superpolynomials for torus knots from evolution induced by cut-and-join operators”, JHEP, 03 (2013), 021, 85 pp., arXiv: ; A. Mironov, A. Morozov, S. Shakirov, A. Sleptsov, “Interplay between MacDonald and Hall–Littlewood expansions of extended torus superpolynomials”, JHEP, 05 (2012), 70, 11 pp., arXiv: ; I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv: ; “DAHA–Jones polynomials of torus knots”, Selecta Math. (N. S.), 22:2 (2016), 1013–1053, arXiv: ; E. Gorsky, A. Oblomkov, J. Rasmussen, “On stable Khovanov homology of torus knots”, Exp. Math., 22:3 (2013), 265–281, arXiv: ; E. Gorsky, A. Negut, “Refined knot invariants and Hilbert schemes”, J. Math. Pures Appl. (9), 104:3 (2015), 403–435, arXiv: ; I. Cherednik, I. Danilenko, “DAHA and iterated torus knots”, Algebr. Geom. Topol., 16:2 (2016), 843–898, arXiv: hep-th/0412243math/05056621105.51171202.24891106.43051201.33391111.61951406.39591206.22261304.33281408.4348 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI | MR

[7] M. Aganagic, A. Klemm, M. Mariño, C. Vafa, “The topological vertex”, Commun. Math. Phys., 254:2 (2005), 425–478, arXiv: ; A. Iqbal, C. Kozcaz, C. Vafa, “The refined topological vertex”, JHEP, 10 (2009), 069, 58 pp. ; S. Gukov, A. Iqbal, C. Kozçaz, C. Vafa, “Link homologies and the refined topological vertex”, Commun. Math. Phys., 298:3 (2010), 757–785, arXiv: ; M. Taki, “Refined topological vertex and instanton counting”, JHEP, 03 (2008), 048, 22 pp., arXiv: ; H. Awata, H. Kanno, “Changing the preferred direction of the refined topological vertex”, J. Geom. Phys., 64 (2013), 91–110, arXiv: ; N. Nekrasov, A. Okounkov, “Membranes and sheaves”, Algebr. Geom., 3:3 (2016), 320–369, arXiv: ; Y. Zenkevich, “Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions”, JHEP, 05 (2015), 131, 22 pp., arXiv: hep-th/03051320705.13680710.17760903.53831404.23231412.8592 | DOI | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | MR | DOI | MR

[8] A. Yu. Morozov, “Zagadki $\beta$-deformatsii”, TMF, 173:1 (2012), 104–126 | DOI | DOI

[9] E. Gorsky, S. Gukov, M. Stosic, Quadruply-graded colored homology of knots, arXiv: 1304.3481

[10] S. Arthamonov, Sh. Shakirov, Refined Chern–Simons theory in genus two, arXiv: 1504.02620

[11] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, “Generalized Kazakov–Migdal–Kontsevich model: group theory aspects”, Internat. J. Modern Phys. A, 10:14 (1995), 2015–2052 ; А. Д. Миронов, А. Ю. Морозов, С. М. Натанзон, “Полный набор операторов разрезания и склейки в теории Гурвица–Концевича”, ТМФ, 166:1 (2011), 3–27, arXiv: ; A. Mironov, A. Morozov, S. Natanzon, “Algebra of differential operators associated with Young diagrams”, J. Geom. Phys., 62:2 (2012), 148–155, arXiv: ; А. Д. Миронов, А. Ю. Морозов, А. В. Слепцов, “Разложение по родам для полиномов ХОМФЛИ”, ТМФ, 177:2 (2013), 179–221, arXiv: ; A. Mironov, A. Morozov, A. Sleptsov, “On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions”, Eur. Phys. J. C, 73:7 (2013), 2492, arXiv: ; A. Mironov, A. Morozov, A. Sleptsov, A. Smirnov, “On genus expansion of superpolynomials”, Nucl. Phys. B, 889 (2014), 757–777, arXiv: ; A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, “Integrability of Hurwitz partition functions”, J. Phys. A: Math.Theor., 45:4 (2012), 045209, 10 pp., arXiv: ; “On KP-integrable Hurwitz functions”, JHEP, 11 (2014), 080, 30 pp., arXiv: 0904.42271012.04331303.10151304.74991310.76221103.41001405.1395 | DOI | MR | DOI | DOI | DOI | MR | Zbl | DOI | Zbl | DOI | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[12] A. Yu. Morozov, “Teoriya strun – chto eto takoe?”, UFN, 162:8, 83–175 ; “Интегрируемость и матричные модели”, УФН, 164:1 (1994), 3–62, arXiv: ; A. Morozov, “Matrix models as integrable systems”, Particles and Fields (Banff, Canada, 16–24 August, 1994), CRM Series in Mathematical Physics, eds. G. W. Semenoff, L. Vinet, Springer, New York, 1999, 127–210, arXiv: ; “Challenges of matrix models”, String Theory: From Gauge Interactions to Cosmology, NATO Science Series II: Mathematics, Physics and Chemistry, 208, eds. L. Baulieu, J. de Boer, B. Pioline, E. Rabinovici, Springer, Dordrecht, 2006, 129–162, arXiv: ; “$2d$ gravity and matrix models. I. $2d$ gravity”, Internat. J. Modern Phys. A, 9:25 (1994), 4355–4405, arXiv: ; А. Д. Миронов, “Матричные модели двумерной гравитации”, ЭЧАЯ, 33:5 (2002), 1051–1145; “Матричные модели и матричные интегралы”, ТМФ, 146:1 (2006), 77–89, arXiv: hep-th/9303139hep-th/9502091hep-th/0502010hep-th/9312212hep-th/0506158 | DOI | DOI | DOI | DOI | MR | DOI | DOI | MR | DOI | DOI | MR | Zbl

[13] A. Alexandrov, A. Mironov, A. Morozov, “Partition functions of matrix models: first special functions of string theory”, Internat. J. Modern Phys. A, 19:24 (2004), 4127–4165, arXiv: ; “Unified description of correlators in non-gaussian phases of hermitian matrix model”, Internat. J. Modern Phys. A, 21:12 (2006), 2481–2517, arXiv: ; “Unified description of correlators in non-Gaussian phases of hermitian matrix model”, Internat. J. Modern Phys. A, 21:12 (2006), 2481–2518, arXiv: ; “Solving Virasoro constraints in matrix models”, Fortsch. Phys., 53:5–6 (2005), 512–521, arXiv: ; “Instantons and merons in matrix models”, Phys. D, 235:1–2 (2007), 126–167, arXiv: ; “BGWM as second constituent of complex matrix model”, JHEP, 12 (2009), 053, 49 pp., arXiv: ; А. С. Александров, А. Д. Миронов, А. Ю. Морозов, “$M$-теория матричных моделей”, ТМФ, 150:2 (2007), 179–192, arXiv: ; L. Chekhov, B. Eynard, N. Orantin, “Free energy topological expansion for the 2-matrix model”, JHEP, 12 (2006), 053, 31 pp., arXiv: ; B. Eynard, N. Orantin, “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys., 1:2 (2007), 347–452, arXiv: ; A. Alexandrov, A. Mironov, A. Morozov, P. Putrov, “Partition functions of matrix models as the first special functions of string theory II: Kontsevich model”, Internat. J. Modern Phys. A, 24:27 (2009), 4939–4998, arXiv: hep-th/0310113hep-th/0412099hep-th/0412099hep-th/0412205hep-th/06082280906.3305hep-th/0605171math-ph/0603003math-ph/07020450811.2825 | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[14] J. C. Baez, J. Dolan, “Categorification”, Higher Category Theory (Northwestern University, Evanston, IL, March 28–30, 1997), Contemporary Mathematics, 230, eds. E. Getzler, M. Kapranov, AMS, Providence, RI, 1998, 1–36, arXiv: ; L. Crane, D. Yetter, “Examples of categorification”, Cah. Topol. Géom. Différ. Catég., 39:1 (1998), 3–25 ; V. Mazorchuk, Lectures on Algebraic Categorification, QGM Master Class Series, EMS, Zürich, 2012 ; A. Savage, Introduction to categorification, arXiv: ; M. Khovanov, V. Mazorchuk, C. Stroppel, “A brief review of abelian categorifications”, Theory Appl. Categ., 22:19 (200), 479–508, arXiv: math.QA/98020291401.6037math.RT/0702746 | MR | MR | DOI | MR | Zbl | MR

[15] E. Witten, “Supersymmetry and Morse theory”, J. Differential Geom., 17:4 (1982), 661–692 ; “Khovanov homology and gauge theory”, Proceedings of the Freedman Fest (Santa Barbara, CA, USA, April 15–17, 2011; Berkeley, CA, USA, June 6–10, 2011), Geometry and Topology Monographs, 18, eds. R. Kirby, V. Krushkal, Z. Wang, Geom. Topol. Publ., Coventry, 2012, 291–308, arXiv: ; A. Kapustin, E. Witten, “Electric-magnetic duality and the geometric Langlands Program”, Commun. Number Theory Phys., 1:1 (2007), 1–236, arXiv: 1108.3103hep-th/0604151 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[16] L. H. Kauffman, “State models and the Jones polynomial”, Topology, 26:3 (1987), 395–407 ; “Invariants of graphs in three-space”, Trans. Amer. Math. Soc., 311:2 (1989), 697–710 ; L. H. Kauffman, P. Vogel, “Link polynomials and a graphical calculus”, J. Knot Theory Ramifications, 1:1 (1992), 59–104 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[17] M. Khovanov, “Patterns in knot cohomology. I”, Exp. Math., 12:3 (2003), 365–374, arXiv: ; “Categorifications of the colored Jones polynomial”, J. Knot Theory Ramifications, 14:1 (2005), 111–130, arXiv: ; “$sl(3)$ link homology”, Algebr. Geom. Topol., 4 (2004), 1045–1081, arXiv: ; “Triply-graded link homology and Hochschild homology of Soergel bimodules”, Internat. J. Math., 18:8 (2007), 869–885, arXiv: ; “Link homology and categorification”, Proceedings of the International Congress of Mathematicians (Madrid, 22–30 August, 2006), v. 2, eds. M. Sanz-Solé, J. Soria, J. L. Varona, J. Verdera, Eur. Math. Soc., Zürich, 989–999, arXiv: ; M. Khovanov, “Categorifications from planar diagrammatics”, Japanese J. Math., 5:2 (2010), 153–181, arXiv: math/0201306math/0302060math/0304375math/0510265math/06053391008.5084 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[18] D. Bar-Natan, “On Khovanov's categorification of the Jones polynomial”, Algebr. Geom. Topol., 2:1 (2002), 337–370, arXiv: ; “Khovanov's homology for tangles and cobordisms”, Geom. Topol., 9:3 (2005), 1443–1499, arXiv: ; D. Bar-Natan, “Fast Khovanov homology computations”, J. Knot Theory Ramifications, 16:3 (2007), 243–255, arXiv: math/0201043math/0410495math/0606318 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[19] V. Dolotin, A. Morozov, “Introduction to Khovanov homologies. I. Unreduced Jones superpolynomial”, JHEP, 01 (2013), 065, 46 pp., arXiv: 1208.4994 | DOI | MR

[20] V. Dolotin, A. Morozov, “Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov–Rozansky invariants”, Nucl. Phys. B, 878 (2014), 12–81, arXiv: 1308.5759 | DOI | MR | Zbl

[21] A. Morozov, And. Morozov, A. Popolitov, “On matrix-model approach to simplified Khovanov–Rozansky calculus”, Phys. Lett. B, 749 (2015), 309–325, arXiv: 1506.07516 | DOI

[22] V. Dolotin, A. Morozov, Introduction to Non-Linear Algebra, World Sci., Singapore, 2007, arXiv: hep-th/0609022 | MR

[23] A. Anokhina, A. Morozov, “Towards $\mathcal R$-matrix construction of Khovanov–Rozansky polynomials. I. Primary $T$-deformation of HOMFLY”, JHEP, 07 (2014), 063, 180 pp., arXiv: 1403.8087 | DOI | MR

[24] A. Morozov, And. Morozov, Ant. Morozov, “On possible existence of HOMFLY polynomials for virtual knots”, Phys. Lett. B, 737 (2014), 48–56, arXiv: ; L. Bishler, A. Morozov, An. Morozov, Ant. Morozov, “Evolution method and HOMFLY polynomials for virtual knots”, Internat. J. Modern Phys. A, 30:14 (2015), 1550074, 39 pp., arXiv: 1407.63191411.2569 | DOI | Zbl | DOI | MR | Zbl

[25] N. Yu. Reshetikhin, V. G. Turaev, “Ribbon graphs and their invaraints derived from quantum groups”, Commun. Math. Phys., 127:1 (1990), 1–26 | DOI | MR | Zbl

[26] R. K. Kaul, T. R. Govindarajan, “Three-dimensional Chern–Simons theory as a theory of knots and links”, Nucl. Phys. B, 380:1–2 (1992), 293–333, arXiv: ; “Three-dimensional Chern–Simons theory as a theory of knots and links: (II). Multicoloured links”, 393:1–2 (1993), 392–412 ; P. Rama Devi, T. R. Govindarajan, R. K. Kaul, “Three-dimensional Chern–Simons theory as a theory of knots and links. (III). Compact semi-simple group”, Nucl. Phys. B, 402:1–2 (1993), 548–566, arXiv: ; “Knot invariants from rational conformal field theories”, Nucl. Phys. B, 422:1–2 (1994), 291–306, arXiv: ; “Representations of composite braids and invariants for mutant knots and links in Chern–Simons field theories”, Modern Phys. Lett. A, 10:22 (1995), 1635–1658, arXiv: ; Zodinmawia, P. Ramadevi, “$SU(N)$ quantum Racah coefficients and non-torus links”, Nucl. Phys. B, 870:1 (2013), 205–242, arXiv: ; Reformulated invariants for non-torus knots and links, arXiv: ; D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, “Colored knot polynomials for arbitrary Pretzel knots and links”, Phys. Lett. B, 743 (2015), 71–74, arXiv: ; A. Mironov, A. Morozov, A. Sleptsov, “Colored HOMFLY polynomials for the pretzel knots and links”, JHEP, 07 (2015), 069, 34 pp., arXiv: ; D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, “Knot invariants from Virasoro related representation and Pretzel knots”, Nucl. Phys. B, 899 (2015), 194–228, arXiv: ; S. Nawata, P. Ramadevi, V. K. Singh, Colored HOMFLY polynomials that distinguish mutant knots, arXiv: ; A. Mironov, A. Morozov, And. Morozov, P. Ramadevi, V. K. Singh, JHEP, 07 (2015), 109, 68 pp., arXiv: hep-th/9111063hep-th/9212110hep-th/9312215hep-th/94120841107.39181209.13461412.26161412.84321502.026211504.003641504.00371 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | DOI | MR | Zbl | DOI

[27] A. Morozov, A. Smirnov, “Chern–Simons teory in the temporal gauge and knot invariants through the universal quantum $R$-matrix”, Nucl. Phys. B, 835:3 (2010), 284–313, arXiv: ; A. Smirnov, “Notes on Chern–Simons theory in the temporal gauge”, Proceedings of 47th International School of Subnuclear Physics (Erice, Italy, 29 August –7 September, 2009 Sicily), ed. A. Zichichi, World Sci., Singapore, 2012, 489–498, arXiv: ; A. Mironov, A. Morozov, And. Morozov, “Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid”, JHEP, 03 (2012), 034, 33 pp., arXiv: ; “Character expansion for HOMFLY polynomials I. Integrability and difference equations”, Strings, Gauge Fields, and the Geometry Behind (The Legacy of Maximilian Kreuzer), eds. A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, E. Scheidegger, World Sci., Singapore, 2013, 101–118, arXiv: ; A. Mironov, A. Morozov, And. Morozov, “Evolution method and ‘differential hierarchy’ of colored knot polynomials”, AIP Conf. Proc., 1562:1 (2013), 123–155, arXiv: ; A. Anokhina, A. Mironov, A. Morozov, And. Morozov, “Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux”, Adv. High Energy Phys., 2013, 931830, 12 pp., arXiv: ; С. Б. Артамонов, А. Д. Миронов, А. Ю. Морозов, “Иерархия дифференциалов и дополнительная градуировка полиномов узлов”, ТМФ, 179:2 (2014), 147–188, arXiv: ; А. С. Анохина, А. А. Морозов, “Процедура каблирования для раскрашенных полиномов ХОМФЛИ”, ТМФ, 178:1 (2014), 3–68, arXiv: ; Ya. Kononov, A. Morozov, “On the defect and stability of differential expansion”, Письма в ЖЭТФ, 101:12 (2015), 931–934, arXiv: ; “Factorization of colored knot polynomials at roots of unity”, Phys. Lett. B, 747 (2015), 500–510, arXiv: ; A. Mironov, A. Morozov, “Towards effective topological field theory for knots”, Nucl. Phys. B, 899 (2015), 395–413, arXiv: 1001.20030910.50111112.26541112.57541306.31971304.14861306.56821307.22161504.071461505.061701506.00339 | DOI | MR | Zbl | DOI | MR | MR | DOI | MR | Zbl | DOI | DOI | DOI | DOI | MR | Zbl | DOI | DOI | DOI | MR

[28] O. T. Dasbach, A. M. Lowrance, “A Turaev surface approach to Khovanov homology”, Quantum Topol., 5:4 (2014), 425–486, arXiv: 1107.2344 | DOI | MR | Zbl

[29] A. Brini, B. Eynard, M. Mariño, “Torus knots and mirror symmetry”, Ann. Henri Poincaré, 13:8 (2012), 1873–1910, arXiv: ; A. Aleksandrov, A. D. Mironov, A. Morozov, A. A. Morozov, “Towards matrix model representation of HOMFLY polynomials”, Письма в ЖЭТФ, 100:4 (2014), 297–304 ; J. Gu, A. Klemm, M. Mariño, J. Reuter, “Exact solutions to quantum spectral curves by topological string theory”, JHEP, 10 (2015), 025, 68 pp., arXiv: 1105.20121506.09176 | DOI | MR | Zbl | DOI | DOI | DOI | MR

[30] P. Cvitanović, Group Theory: Birdtracks, Lie's, and Exceptional Groups, Princeton Univ. Press, Princeton, NJ, 2008 | MR | Zbl