Towards ℛ-matrix construction of Khovanov-Rozansky polynomials I. Primary T-deformation of HOMFLY

A. Anokhina1, А. Морозов1
1ITEP, B. Cheremushkinskaya 25, Moscow, Russia

Tóm tắt

Từ khóa


Tài liệu tham khảo

V. Dolotin and A. Morozov, Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants, Nucl. Phys. B 878 (2014) 12 [ arXiv:1308.5759 ] [ INSPIRE ].

S. Chern and J. Simons, Some cohomology classes in principal fiber bundles and their application to Riemannian geometry, Proc. Nat. Acad. Sci. 68 (1971) 791.

S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals Math. 99 (1974) 48 [ INSPIRE ].

A.S. Schwarz, New topological invariants arising in the theory of quantized fields, Baku Topol. Conf., (1987).

E. Witten, Quantum Field Theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [ INSPIRE ].

M.Atiyah, The geometry and physics of knots, CUP, Cambridge U.K. (1990).

A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [ INSPIRE ].

V.G. Knizhnik and A.B. Zamolodchikov, Current algebra and Wess-Zumino model in two-dimensions, Nucl. Phys. B 247 (1984) 83 [ INSPIRE ].

Al.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [ INSPIRE ].

Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088 [Teor. Mat. Fiz. 73 (1987) 103].

Al. Zamolodchikov and A. Zamolodchikov, Conformal Field Theory and critical phenomena in 2d systems (in Russian), (2009).

V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [ INSPIRE ].

G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field theory, Commun. Math. Phys. 123 (1989) 177 [ INSPIRE ].

A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [ INSPIRE ].

P.H. Ginsparg, Applied Conformal Field Theory, hep-th/9108028 [ INSPIRE ].

P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer, Germany (1997).

J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [ hep-th/0104158 ] [ INSPIRE ].

A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, Theor. Math. Phys. 165 (2010) 1662 [Teor. Mat. Fiz. 165 (2010) 503] [ arXiv:0908.2064 ] [ INSPIRE ].

V.G. Knizhnik and A.Y. Morozov, Renormalization of topological charge, JETP Lett. 39 (1984) 240 [ INSPIRE ].

H. Levine and S.B. Libby, Renormalization of the θ angle, the quantum hall effect and the strong CP problem, Phys. Lett. B 150 (1985) 182 [ INSPIRE ].

N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [ hep-th/9407087 ] [ INSPIRE ].

N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [ hep-th/9408099 ] [ INSPIRE ].

A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [ hep-th/9505035 ] [ INSPIRE ].

R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [ hep-th/9510101 ] [ INSPIRE ].

D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [ hep-th/9608163 ] [ INSPIRE ].

E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [ hep-th/9610234 ] [ INSPIRE ].

A. Marshakov, M. Martellini and A. Morozov, Insights and puzzles from branes: 4D SUSY Yang-Mills from 6D models, Phys. Lett. B 418 (1998) 294 [ hep-th/9706050 ] [ INSPIRE ].

A. Losev, G.W. Moore, N. Nekrasov and S. Shatashvili, Four-dimensional avatars of two-dimensional RCFT, Nucl. Phys. Proc. Suppl. 46 (1996) 130 [ hep-th/9509151 ] [ INSPIRE ].

A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [ hep-th/9711108 ] [ INSPIRE ].

A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [ INSPIRE ].

G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [ hep-th/9712241 ] [ INSPIRE ].

G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [ hep-th/9803265 ] [ INSPIRE ].

N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [ hep-th/0206161 ] [ INSPIRE ].

N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [ INSPIRE ].

R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [ hep-th/0208176 ] [ INSPIRE ].

S. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033 [ hep-th/0408066 ] [ INSPIRE ].

S. Shadchin, Status report on the instanton counting, SIGMA 2 (2006) 008 [ hep-th/0601167 ] [ INSPIRE ].

V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [ arXiv:0712.2824 ] [ INSPIRE ].

V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [ arXiv:0906.0638 ] [ INSPIRE ].

N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [ INSPIRE ].

N. Nekrasov and E. Witten, The omega deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [ arXiv:1002.0888 ] [ INSPIRE ].

D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [ arXiv:0904.2715 ] [ INSPIRE ].

L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [ arXiv:0906.3219 ] [ INSPIRE ].

D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [ arXiv:0807.4723 ] [ INSPIRE ].

D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [ INSPIRE ].

D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS states, arXiv:1006.0146 [ INSPIRE ].

D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing in coupled 2d-4d systems, arXiv:1103.2598 [ INSPIRE ].

D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré 14 (2013) 1643 [ arXiv:1204.4824 ] [ INSPIRE ].

N. Wyllard, A N −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [ arXiv:0907.2189 ] [ INSPIRE ].

A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [ arXiv:0908.2569 ] [ INSPIRE ].

A. Marshakov, A. Mironov and A. Morozov, Combinatorial expansions of conformal blocks, Theor. Math. Phys. 164 (2010) 831 [ arXiv:0907.3946 ] [ INSPIRE ].

A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [ arXiv:0909.2052 ] [ INSPIRE ].

A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [ arXiv:0909.3338 ] [ INSPIRE ].

A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [ arXiv:1011.4491 ] [ INSPIRE ].

N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [ INSPIRE ].

L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [ arXiv:0909.0945 ] [ INSPIRE ].

N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [ arXiv:0909.1105 ] [ INSPIRE ].

R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [ arXiv:0909.3412 ] [ INSPIRE ].

G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [ arXiv:0909.4031 ] [ INSPIRE ].

H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [ arXiv:0910.4431 ] [ INSPIRE ].

V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, JETP Lett. 90 (2009) 708 [ arXiv:0911.0363 ] [ INSPIRE ].

V. Alba and A. Morozov, Check of AGT relation for conformal blocks on sphere, Nucl. Phys. B 840 (2010) 441 [ arXiv:0912.2535 ] [ INSPIRE ].

L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [ arXiv:0911.2353 ] [ INSPIRE ].

L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for N f = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [ arXiv:1004.1841 ] [ INSPIRE ].

A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [ arXiv:0910.5670 ] [ INSPIRE ].

G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via beta-deformed matrix models, arXiv:1104.4016 [ INSPIRE ].

H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [ arXiv:1105.0357 ] [ INSPIRE ].

R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [ INSPIRE ].

H. Itoyama, K. Maruyoshi and T. Oota, The quiver matrix model and 2d-4d conformal connection, Prog. Theor. Phys. 123 (2010) 957 [ arXiv:0911.4244 ] [ INSPIRE ].

T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [ arXiv:0911.4797 ] [ INSPIRE ].

T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP 07 (2010) 081 [ arXiv:1006.0828 ] [ INSPIRE ].

R. Schiappa and N. Wyllard, An A r threesome: matrix models, 2d CFTs and 4d N = 2 gauge theories, J. Math. Phys. 51 (2010) 082304 [ arXiv:0911.5337 ] [ INSPIRE ].

A. Mironov, A. Morozov and S. Shakirov, Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP 02 (2010) 030 [ arXiv:0911.5721 ] [ INSPIRE ].

A. Mironov, A. Morozov and S. Shakirov, Conformal blocks as Dotsenko-Fateev integral discriminants, Int. J. Mod. Phys. A 25 (2010) 3173 [ arXiv:1001.0563 ] [ INSPIRE ].

A. Mironov, A. Morozov and S. Shakirov, On ‘Dotsenko-Fateev’ representation of the toric conformal blocks, J. Phys. A 44 (2011) 085401 [ arXiv:1010.1734 ] [ INSPIRE ].

A. Mironov, A. Morozov and S. Shakirov, A direct proof of AGT conjecture at β = 1, JHEP 02 (2011) 067 [ arXiv:1012.3137 ] [ INSPIRE ].

H. Itoyama and T. Oota, Method of generating q-expansion coefficients for conformal block and N = 2 Nekrasov function by beta-deformed matrix model, Nucl. Phys. B 838 (2010) 298 [ arXiv:1003.2929 ] [ INSPIRE ].

A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys. B 843 (2011) 534 [ arXiv:1003.5752 ] [ INSPIRE ].

A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys. B 855 (2012) 128 [ arXiv:1105.0948 ] [ INSPIRE ].

H. Zhang and Y. Matsuo, Selberg integral and SU(N) AGT conjecture, JHEP 12 (2011) 106 [ arXiv:1110.5255 ] [ INSPIRE ].

D. Galakhov, A. Mironov and A. Morozov, S-duality and modular transformation as a non-perturbative deformation of the ordinary pq-duality, arXiv:1311.7069 [ INSPIRE ].

S. Mironov, A. Morozov and Y. Zenkevich, Generalized Jack polynomials and the AGT relations for the SU(3) group, JETP Lett. 99 (2014) 109 [ arXiv:1312.5732 ] [ INSPIRE ].

V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [ arXiv:0912.0504 ] [ INSPIRE ].

V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [ arXiv:1012.1312 ] [ INSPIRE ].

A. Belavin and V. Belavin, AGT conjecture and integrable structure of Conformal Field Theory for c = 1, Nucl. Phys. B 850 (2011) 199 [ arXiv:1102.0343 ] [ INSPIRE ].

A.A. Belavin, M.A. Bershtein, B.L. Feigin, A.V. Litvinov and G.M. Tarnopolsky, Instanton moduli spaces and bases in coset Conformal Field Theory, Comm. Math. Phys. 319 (2013) 269 [ arXiv:1111.2803 ] [ INSPIRE ].

S. Kanno, Y. Matsuo and H. Zhang, Virasoro constraint for Nekrasov instanton partition function, JHEP 10 (2012) 097 [ arXiv:1207.5658 ] [ INSPIRE ].

S. Kanno, Y. Matsuo and H. Zhang, Extended conformal symmetry and recursion formulae for Nekrasov partition function, JHEP 08 (2013) 028 [ arXiv:1306.1523 ] [ INSPIRE ].

H. Nakajima, Lectures on Hilbert schemes of points on surfaces, AMS, U.S.A. (1999).

H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365.

H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998) 515.

H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math. 145 (1997) 379388 [ alg-geom/9507012 ].

H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, alg-geom/9610021 .

H. Nakajima, Quiver varieties and finite dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2000) 145 [ math.QA/9912158 ].

A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010) 523 [ math.AG/0411210 ].

W.-P. Li, Z. Qin and W. Wang, The cohomology rings of Hilbert schemes via Jack polynomials, CRM Proc. Lect. Notes 38 (2004) 249258 [ math.AG/0411255 ].

A. Smirnov, On the instanton R-matrix, arXiv:1302.0799 [ INSPIRE ].

S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002) 497 [ math.RT/0104151 ].

S. Fomin and A. Zelevinsky, Cluster algebras IV: coefficients, Composito Math. 143 (2007) 112 [ math.RA/0602259 ].

V.V. Fock and A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 [ math.AG/0311149 ].

S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. part I: cluster complexes, Acta Math. 201 (2008) 83 [ math.RA/0608367 ].

S. Fomin and D. Thurston, Cluster algebras and triangulated surfaces. Part II: Λ lengths, arXiv:1210.5569 .

V.V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, arXiv:1401.1606 [ INSPIRE ].

A. Popolitov, The cluster variety face of quantum groups, arXiv:1403.1834 [ INSPIRE ].

T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [ arXiv:1006.0977 ] [ INSPIRE ].

T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [ arXiv:1108.4389 ] [ INSPIRE ].

T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, arXiv:1112.5179 [ INSPIRE ].

D.V. Galakhov et al., Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation, Theor. Math. Phys. 172 (2012) 939 [ arXiv:1104.2589 ] [ INSPIRE ].

A. Mironov and A. Morozov, Equations on knot polynomials and 3d/5d duality, AIP Conf. Proc. 1483 (2012) 189 [ arXiv:1208.2282 ] [ INSPIRE ].

M. Adams, J.P. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys. 20 (1990) 299 [ INSPIRE ].

E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and g l N , g l M $$ \left(\mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M\right) $$ dualities, math.QA/0510364 .

E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and g l N , g l M $$ \left(\mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M\right) $$ dualities, discrete versus differential, math.QA/0605172 .

E. Mukhin, V. Tarasov and A. Varchenko, A generalization of the Capelli identity, math.QA/0610799 .

A. Mironov, A. Morozov, Y. Zenkevich and A. Zotov, Spectral duality in integrable systems from AGT conjecture, JETP Lett. 97 (2013) 45 [ arXiv:1204.0913 ] [ INSPIRE ].

A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral duality between Heisenberg chain and Gaudin model, Lett. Math. Phys. 103 (2013) 299 [ arXiv:1206.6349 ] [ INSPIRE ].

A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP 12 (2013) 034 [ arXiv:1307.1502 ] [ INSPIRE ].

B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [ INSPIRE ].

B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U q (sl(2, R)), Commun. Math. Phys. 224 (2001) 613 [ math.QA/0007097 ] [ INSPIRE ].

J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, hep-th/0308031 [ INSPIRE ].

D. Galakhov, A. Mironov and A. Morozov, S-duality as a beta-deformed Fourier transform, JHEP 08 (2012) 067 [ arXiv:1205.4998 ] [ INSPIRE ].

D. Galakhov, A. Mironov and A. Morozov, S-duality and modular transformation as a non-perturbative deformation of the ordinary pq-duality, arXiv:1311.7069 [ INSPIRE ].

M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Deformed N = 2 theories, generalized recursion relations and S-duality, JHEP 04 (2013) 039 [ arXiv:1302.0686 ] [ INSPIRE ].

M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Modular anomaly equation, heat kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [ arXiv:1307.6648 ] [ INSPIRE ].

N. Nemkov, S-duality as Fourier transform for arbitrary ϵ 1 , ϵ 2, arXiv:1307.0773 [ INSPIRE ].

N. Iorgov, O. Lisovyy and Y. Tykhyy, Painlevé VI connection problem and monodromy of c = 1 conformal blocks, JHEP 12 (2013) 029 [ arXiv:1308.4092 ] [ INSPIRE ].

M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville triality, arXiv:1309.1687 [ INSPIRE ].

M. Aganagic, N. Haouzi and S. Shakirov, A n -triality, arXiv:1403.3657 [ INSPIRE ].

J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275.

J.H. Conway, Algebraic properties, in Computational problems in abstract algebra, J. Leech ed., Proc. Conf. Oxford, 1967, Pergamon Press, Oxford U.K. and New York U.S.A. (1970), pg. 329.

V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1.

V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985) 103.

V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals Math. 126 (1987) 335 [ INSPIRE ].

L. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395.

P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millet and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239.

J.H. Przytycki and K.P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987) 115.

E. Witten, Two lectures on the Jones polynomial and Khovanov homology, arXiv:1401.6996 [ INSPIRE ].

S.A. Cherkis, Octonions, monopoles and knots, arXiv:1403.6836 [ INSPIRE ].

M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [ math.QA/9908171 ].

D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Alg. Geom. Topol. 2 (2002) 337 [ math.QA/0201043 ].

M. Khovanov, Patterns in knot cohomology I, Exper. Math. 12 (2003) 365 [ math.QA/0201306 ].

M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14 (2005) 111 [ math.QA/0302060 ].

M. Khovanov, sl 3 $$ \mathfrak{sl}(3) $$ link homology, Algebr. Geom. Topol. 4 (2004) 1045 [ math.QA/0304375 ].

M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Int. J. Math. 18 (2007) 869 [ math.GT/0510265 ].

M. Khovanov, Link homology and categorification, math.QA/0605339 .

M. Khovanov, Categorifications from planar diagrammatics, arXiv:1008.5084 .

M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 [ math.QA/0401268 ].

M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom. Topol. 12 (2008) 1387 [ math.QA/0505056 ].

M. Khovanov and L. Rozansky, Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial, math.QA/0701333 .

A.Shumakovitch, Torsion of the Khovanov homology, math.GT/0405474 .

D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 [ math.GT/0410495 ].

D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243 [ math.GT/0606318 ].

L. Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365 [ math.GT/0407071 ].

S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [ hep-th/0412243 ] [ INSPIRE ].

J. Rasmussen, Khovanov-Rozansky homology of two-bridge knots and links, Duke Math. J. 136 (2007) 551 [ math.GT/0508510 ].

J. Rasmussen, Some differentials on Khovanov-Rozansky homology, math.GT/0607544 .

S. Gukov and J. Walcher, Matrix factorizations and Kauffman homology, hep-th/0512298 [ INSPIRE ].

E. Wagner, Sur l’homologie de Khovanov-Rozansky des graphes et des entrelacs (in French), Université Louis Pasteur, Strasbourg France (2007).

M. Stosic, Homological thickness and stability of torus knots, Algebr. Geom. Topol. 7 (2007) 261 [ math.GT/0511532 ].

M. Stosic, Khovanov homology of links and graphs, math.QA/0605579 .

M. Stosic, Homology of torus links, Topology Appl. 156 (2009) 533 [ math.QA/0606656 ].

P. Turner, A spectral sequence for Khovanov homology with an application to (3, q)-torus links, Algebr. Geom. Topol. 8 (2008) 869 [ math.GT/0606369 ].

P. Turner, Five lectures on Khovanov homology, math.GT/0606464 .

E. Gorsky, q, t-Catalan numbers and knot homology, Contemp. Math. 566, AMS, Providence U.S.A. (2012), pg. 213 [ arXiv:1003.0916 ].

E. Gorsky, Arc spaces and DAHA representations, Selecta Math. 19 (2013) 125 [ arXiv:1110.1674 ].

E. Gorsky and M. Mazin, Compactified Jacobians and q, t-Catalan numbers, I, J. Combinat. Theor. A 120 (2013) 49 [ arXiv:1105.1151 ].

E. Gorsky and M. Mazin, Compactified Jacobians and q, t-Catalan numbers, II, J. Alg. Combinat. 39 (2014) 153 [ arXiv:1204.5448 ].

L. Rozansky, An infinite torus braid yields a categorified Jones-Wenzl projector, arXiv:1005.3266 .

B. Webster, Knot invariants and higher representation theory II: the categorification of quantum knot invariants, arXiv:1005.4559 .

N. Carqueville and D. Murfet, Computing Khovanov-Rozansky homology and defect fusion, Topology 14 (2014) 489 [ arXiv:1108.1081 ] [ INSPIRE ].

H. Wu, Colored sl N $$ \mathfrak{sl}(N) $$ link homology via matrix factorizations, arXiv:1110.2076 .

A. Oblomkov, J. Rasmussen and V. Shende, The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link, arXiv:1201.2115 [ INSPIRE ].

E. Gorsky, A. Oblomkov and J. Rasmussen, On stable Khovanov homology of torus knots, Exper. Math. 22 (2013) 265 [ arXiv:1206.2226 ].

N. Carqueville and D. Murfet, Adjunctions and defects in Landau-Ginzburg models, arXiv:1208.1481 [ INSPIRE ].

N. Carqueville and D. Murfet, A toolkit for defect computations in Landau-Ginzburg models, arXiv:1303.1389 [ INSPIRE ].

V. Dolotin and A. Morozov, Introduction to Khovanov homologies. I. Unreduced Jones superpolynomial, JHEP 01 (2013) 065 [ arXiv:1208.4994 ] [ INSPIRE ].

V. Dolotin and A. Morozov, Introduction to Khovanov homologies. II. Reduced Jones superpolynomials, J. Phys. Conf. Ser. 411 (2013) 012013 [ arXiv:1209.5109 ] [ INSPIRE ].

R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [ hep-th/9811131 ] [ INSPIRE ].

H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [ hep-th/9912123 ] [ INSPIRE ].

N.M. Dunfield, S. Gukov and J. Rasmussen, The superpolynomial for knot homologies, Exper. Math. 15 (2006) 129 [ math.GT/0505662 ] [ INSPIRE ].

I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhauser, Boston U.S.A. (1994).

V. Dolotin and A. Morozov, Introduction to non-linear algebra, World Scientific, Singapore (2007) [ hep-th/0609022 ] [ INSPIRE ].

A. Morozov and S. Shakirov, New and old results in resultant theory, arXiv:0911.5278 [ INSPIRE ].

N. Reshetikhin, Invariants of tangles 1, LOMI preprint E-4-87, Russia (1987), http://math.berkeley.edu/~reshetik/Publications/QGInv-1-1987.pdf .

N. Reshetikhin, Invariants of tangles 2, LOMI preprint, E-17-87, Russia (1987), http://math.berkeley.edu/~reshetik/Publications/QGInv-2-1987.pdf .

V. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988) 527 [ INSPIRE ].

E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons holonomies and the appearance of quantum groups, in Proceedings, Quantum groups, Clausthal, (1989), pg. 307 [Phys. Lett. B 235 (1990) 275] [ INSPIRE ].

N.Y. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [ INSPIRE ].

A. Morozov and A. Rosly, unpublished, (1991).

R.K. Kaul and T.R. Govindarajan, Three-dimensional Chern-Simons theory as a theory of knots and links, Nucl. Phys. B 380 (1992) 293 [ hep-th/9111063 ] [ INSPIRE ].

P. Rama Devi, T.R. Govindarajan and R.K. Kaul, Three-dimensional Chern-Simons theory as a theory of knots and links. 3. Compact semisimple group, Nucl. Phys. B 402 (1993) 548 [ hep-th/9212110 ] [ INSPIRE ].

P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Knot invariants from rational conformal field theories, Nucl. Phys. B 422 (1994) 291 [ hep-th/9312215 ] [ INSPIRE ].

P. Ramadevi and T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487 [ hep-th/0009188 ] [ INSPIRE ].

A. Morozov and A. Smirnov, Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix, Nucl. Phys. B 835 (2010) 284 [ arXiv:1001.2003 ] [ INSPIRE ].

M. Aganagic and S. Shakirov, Knot homology from refined Chern-Simons theory, arXiv:1105.5117 [ INSPIRE ].

M. Aganagic and S. Shakirov, Refined Chern-Simons theory and knot homology, arXiv:1202.2489 [ INSPIRE ].

P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, Superpolynomials for toric knots from evolution induced by cut-and-join operators, JHEP 03 (2013) 021 [ arXiv:1106.4305 ] [ INSPIRE ].

I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [ INSPIRE ].

A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arXiv:1112.5754 [ INSPIRE ].

A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid, JHEP 03 (2012) 034 [ arXiv:1112.2654 ] [ INSPIRE ].

H. Itoyama, A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. III. All 3-strand braids in the first symmetric representation, Int. J. Mod. Phys. A 27 (2012) 1250099 [ arXiv:1204.4785 ] [ INSPIRE ].

H. Itoyama, A. Mironov, A. Morozov and A. Morozov, Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations, Int. J. Mod. Phys. A 28 (2013) 1340009 [ arXiv:1209.6304 ] [ INSPIRE ].

A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids, Nucl. Phys. B 868 (2013) 271 [ arXiv:1207.0279 ] [ INSPIRE ].

S. Gukov and M. Stosic, Homological algebra of knots and BPS states, arXiv:1112.0030 [ INSPIRE ].

P. Dunin-Barkowski, A. Sleptsov and A. Smirnov, Kontsevich integral for knots and Vassiliev invariants, Int. J. Mod. Phys. A 28 (2013) 1330025 [ arXiv:1112.5406 ] [ INSPIRE ].

P. Dunin-Barkowski, A. Sleptsov and A. Smirnov, Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N), J. Phys. A 45 (2012) 385204 [ arXiv:1201.0025 ] [ INSPIRE ].

A. Mironov, A. Morozov, S. Shakirov and A. Sleptsov, Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials, JHEP 05 (2012) 070 [ arXiv:1201.3339 ] [ INSPIRE ].

A. Mironov, A. Morozov and S. Shakirov, Torus HOMFLY as the Hall-Littlewood polynomials, J. Phys. A 45 (2012) 355202 [ arXiv:1203.0667 ] [ INSPIRE ].

S. Shakirov, Colored knot amplitudes and Hall-Littlewood polynomials, arXiv:1308.3838 [ INSPIRE ].

H. Itoyama, A. Mironov, A. Morozov and A. Morozov, HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations, JHEP 07 (2012) 131 [ arXiv:1203.5978 ] [ INSPIRE ].

Zodinmawia and P. Ramadevi, SU(N) quantum Racah coefficients & non-torus links, Nucl. Phys. B 870 (2013) 205 [ arXiv:1107.3918 ] [ INSPIRE ].

Zodinmawia and P. Ramadevi, Reformulated invariants for non-torus knots and links, arXiv:1209.1346 [ INSPIRE ].

S. Nawata, P. Ramadevi and Zodinmawia, Multiplicity-free quantum 6j-symbols for U q s l N $$ \left(\mathfrak{s}{\mathfrak{l}}_N\right) $$ , Lett. Math. Phys. 103 (2013) 1389 [ arXiv:1302.5143 ] [ INSPIRE ].

S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theor. 22 (2013) 1350078 [ arXiv:1302.5144 ] [ INSPIRE ].

S. Nawata, P. Ramadevi and P. Zodinmawia, Colored Kauffman homology and super-A-polynomials, JHEP 01 (2014) 126 [ arXiv:1310.2240 ] [ INSPIRE ].

E. Gorsky, A. Oblomkov, J. Rasmussen and V. Shende, Torus knots and the rational DAHA, arXiv:1207.4523 [ INSPIRE ].

E. Gorsky and A. Negut, Refined knot invariants and Hilbert schemes, arXiv:1304.3328 [ INSPIRE ].

A. Mironov and A. Morozov, Equations on knot polynomials and 3d/5d duality, AIP Conf. Proc. 1483 (2012) 189 [ arXiv:1208.2282 ] [ INSPIRE ].

H. Fuji, S. Gukov, M. Stosic and P. Sulkowski, 3d analogs of Argyres-Douglas theories and knot homologies, JHEP 01 (2013) 175 [ arXiv:1209.1416 ] [ INSPIRE ].

S. Nawata, P. Ramadevi, Zodinmawia and X. Sun, Super-A-polynomials for twist knots, JHEP 11 (2012) 157 [ arXiv:1209.1409 ] [ INSPIRE ].

A. Morozov, Special colored superpolynomials and their representation-dependence, JHEP 12 (2012) 116 [ arXiv:1208.3544 ] [ INSPIRE ].

A. Morozov, The first-order deviation of superpolynomial in an arbitrary representation from the special polynomial, JETP Lett. 97 (2013) 171 [ arXiv:1211.4596 ] [ INSPIRE ].

A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Adv. High Energy Phys. 2013 (2013) 931830 [ arXiv:1304.1486 ] [ INSPIRE ].

A. Anokhina and A. Morozov, Cabling procedure for the colored HOMFLY polynomials, Teor. Mat. Fiz. 178 (2014) 3 [Teor. Mat. Fiz. 178 (2014) 3] [ arXiv:1307.2216 ] [ INSPIRE ].

A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Knot polynomials in the first non-symmetric representation, Nucl. Phys. B 882 (2014) 171 [ arXiv:1211.6375 ] [ INSPIRE ].

A. Mironov, A. Morozov and A. Sleptsov, Genus expansion of HOMFLY polynomials, Theor. Math. Phys. 177 (2013) 1435 [Teor. Mat. Fiz. 177 (2013) 179] [ arXiv:1303.1015 ] [ INSPIRE ].

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

A. Mironov, A. Morozov and A. Morozov, Evolution method and “differential hierarchy” of colored knot polynomials, AIP Conf. Proc. 1562 (2013) 123 [ arXiv:1306.3197 ] [ INSPIRE ].

S. Arthamonov, A. Mironov and A. Morozov, Differential hierarchy and additional grading of knot polynomials, Theor. Math. Phys. 179 (2014) 509 [ arXiv:1306.5682 ] [ INSPIRE ].

M. Aganagic, T. Ekholm, L. Ng and C. Vafa, Topological strings, D-model and knot contact homology, arXiv:1304.5778 [ INSPIRE ].

S. Arthamonov, A. Mironov, A. Morozov and A. Morozov, Link polynomial calculus and the AENV conjecture, JHEP 04 (2014) 156 [ arXiv:1309.7984 ] [ INSPIRE ].

Q. Chen and N. Reshetikhin, Recursion formulas for HOMFLY and Kauffman invariants, arXiv:1401.1927 .

Q. Chen, K. Liu, P. Peng and S. Zhu, Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials, arXiv:1402.3571 [ INSPIRE ].

I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, New York U.S.A. (1995).

I. Cherednik, Double affine Hecke algebras, Cambridge University Press, Cambridge U.K. (2005).

A. Morozov, Challenges of β-deformation, Theor. Math. Phys. 173 (2012) 1417 [Teor. Mat. Fiz. 173 (2012) 104] [ arXiv:1201.4595 ] [ INSPIRE ].

L. Kauffman, State models and the Jones polynomials, Topology 26 (1987) 395.

L. Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697.

L. Kauffman and P. Vogel, Link polynomials and a graphical calculus, J. Knot Theory Ramifications 1 (1992) 59.

The knot atlas webpage, http://katlas.org/ .

I. Danilenko, Khovanov-Rozansky homologies and cabling, arXiv:1405.0884 [ INSPIRE ].

Table of link invariants webpage, http://www.indiana.edu/~linkinfo/ .

M. Rosso and V.F.R. Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theory Ramifications 2 (1993) 97.

X.-S. Lin and H. Zheng, On the Hecke algebras and the colored HOMFLY polynomial, Trans. Amer. Math. Soc. 362 (2010) 1 [ math.QA/0601267 ].

S. Stevan, Chern-Simons invariants of torus links, Annales Henri Poincaré 11 (2010) 1201 [ arXiv:1003.2861 ] [ INSPIRE ].

A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [ arXiv:1105.2012 ] [ INSPIRE ].

S. Gukov, A. Iqbal, C. Kozcaz and C. Vafa, Link homologies and the refined topological vertex, Commun. Math. Phys. 298 (2010) 757 [ arXiv:0705.1368 ] [ INSPIRE ].

A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, On genus expansion of superpolynomials, arXiv:1310.7622 [ INSPIRE ].