zbMATH — the first resource for mathematics

Superpolynomials for torus knots from evolution induced by cut-and-join operators. (English) Zbl 1342.57004
Summary: The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum \(R\)-matrix and ends up with a trivially-looking split \(W\) representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of ‘superpolynomials’, much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial \(\mathcal P^{[m,km\pm 1]}_{[1]}\) for arbitrary \(m\) and \(k\). The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T45 Topological field theories in quantum mechanics
Full Text: DOI arXiv
[1] Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121, 351, (1989)
[2] E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE].
[3] E. Witten, Fivebranes and Knots, arXiv:1101.3216 [INSPIRE].
[4] Dimofte, T.; Gukov, S.; Hollands, L., Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys., 98, 225, (2011)
[5] Aganagic, M.; Mariño, M.; Vafa, C., All loop topological string amplitudes from Chern-Simons theory, Commun. Math. Phys., 247, 467, (2004)
[6] Aganagic, M.; Klemm, A.; Mariño, M.; Vafa, C., The topological vertex, Commun. Math. Phys., 254, 425, (2005)
[7] Iqbal, A.; Kozcaz, C.; Vafa, C., The refined topological vertex, JHEP, 10, 069, (2009)
[8] Awata, H.; Kanno, H., Refined BPS state counting from nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys., A 24, 2253, (2009)
[9] I.G. Macdonald, Symmetric functions and Hall polynomials, Second Edition, Oxford University Press, 1995.
[10] Ruijsenaars, SNM; Schneider, H., A new class of integrable systems and its relation to solitons, Annals Phys., 170, 370, (1986)
[11] Ruijsenaars, SNM, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys., 110, 191, (1987)
[12] Ruijsenaars, SNM, Action angle maps and scattering theory for some finite dimensional integrable systems. 1. the pure soliton case, Commun. Math. Phys., 115, 127, (1988)
[13] Khovanov, M.; Rozansky, L., Matrix factorizations and link homology, Fund. Math., 199, 1, (2008)
[14] Khovanov, M.; Rozansky, L., Matrix factorizations and link homology II, Geom. Topol., 12, 1387, (2008)
[15] Gukov, S.; Schwarz, AS; Vafa, C., Khovanov-Rozansky homology and topological strings, Lett. Math. Phys., 74, 53, (2005)
[16] N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for knot homologies, math/0505662 [INSPIRE].
[17] Gukov, S.; Iqbal, A.; Kozcaz, C.; Vafa, C., Link homologies and the refined topological vertex, Commun. Math. Phys., 298, 757, (2010)
[18] Alexander, JW, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30, 275, (1928)
[19] J.H. Conway, An Enumeration of Knots and Links, and Some of Their Algebraic Properties, in Proc. Conf. Computational Problems in Abstract Algebra, Oxford, 1967, J. Leech ed., Pergamon Press, Oxford-New York, 329-358, 1970.
[20] Jones, VFR, Index for subfactors, Invent. Math., 72, 1, (1983)
[21] Jones, VFR, A polynomial invariant for links via von Neumann algebras, Bull. AMS, 12, 103, (1985)
[22] Jones, V., Hecke algebra representations of braid groups and link polynomials, Annals Math., 126, 335, (1987)
[23] Kauffman, L., State models and the Jones polynomial, Topology, 26, 395, (1987)
[24] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, WBR; Millet, K.; Ocneanu, A., A new polynomial invariant of knots and links, Bull. AMS, 12, 239, (1985)
[25] Przytycki, JH; Traczyk, KP, Invariants of Conway type, Kobe J. Math., 4, 115, (1987)
[26] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, arXiv:1103.5628.
[27] J.E. Andersen et al., Problems on invariants of knots and 3-manifolds, math/0406190.
[28] Alday, LF; Gaiotto, D.; Tachikawa, Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., 91, 167, (2010)
[29] Wyllard, N., A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP, 11, 002, (2009)
[30] Mironov, A.; Morozov, A., The power of Nekrasov functions, Phys. Lett., B 680, 188, (2009)
[31] Mironov, A.; Morozov, A., On AGT relation in the case of U(3), Nucl. Phys., B 825, 1, (2010)
[32] Terashima, Y.; Yamazaki, M., SL(2, \( \mathbb{R} \)) Chern-Simons, Liouville and gauge theory on duality walls, JHEP, 08, 135, (2011)
[33] Galakhov, D.; Mironov, A.; Morozov, A.; Smirnov, A.; Mironov, A.; etal., Three-dimensional extensions of the Alday-Gaiotto-tachikawa relation, Theor. Math. Phys., 172, 939, (2012)
[34] R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [INSPIRE].
[35] Itoyama, H.; Maruyoshi, K.; Oota, T., The quiver matrix model and 2d-4d conformal connection, Prog. Theor. Phys., 123, 957, (2010)
[36] Eguchi, T.; Maruyoshi, K., Penner type matrix model and Seiberg-Witten theory, JHEP, 02, 022, (2010)
[37] Eguchi, T.; Maruyoshi, K., Seiberg-Witten theory, matrix model and AGT relation, JHEP, 07, 081, (2010)
[38] Schiappa, R.; Wyllard, N., An A(r) threesome: matrix models, 2d CFTs and 4d N = 2 gauge theories, J. Math. Phys., 51, 082304, (2010)
[39] Mironov, A.; Morozov, A.; Shakirov, S., Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP, 02, 030, (2010)
[40] Mironov, A.; Morozov, A.; Shakirov, S., Conformal blocks as dotsenko-fateev integral discriminants, Int. J. Mod. Phys., A 25, 3173, (2010)
[41] M. Aganagic and S. Shakirov, Knot Homology from Refined Chern-Simons Theory, arXiv:1105.5117 [INSPIRE].
[42] E. Guadagnini, M. Martellini and M. Mintchev, Quantum groups, in proceedings of Clausthal 1989, pg. 307-317.
[43] Guadagnini, E.; Martellini, M.; Mintchev, M., Chern-Simons holonomies and the appearance of quantum groups, Phys. Lett., B 235, 275, (1990)
[44] Reshetikhin, NY; Turaev, V., Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., 127, 1, (1990)
[45] Morozov, A.; Smirnov, A., Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix, Nucl. Phys., B 835, 284, (2010)
[46] A. Smirnov, Notes on Chern-Simons Theory in the Temporal Gauge, Proceedings of International School of Subnuclar Phys. in Erice, Italy, 2009 [arXiv:0910.5011] [INSPIRE].
[47] Rosso, M.; Jones, VFR, On the invariants of torus knots derived from quantum groups, J. Knot Theor. Ramif., 2, 97, (1993)
[48] Lin, X-S; Zheng, H., On the Hecke algebras and the colored HOMFLY polynomial, Trans. Amer. Math. Soc., 362, 1, (2010)
[49] Stevan, S., Chern-Simons invariants of torus links, Annales Henri Poincaré, 11, 1201, (2010)
[50] Brini, A.; Eynard, B.; Mariño, M., Torus knots and mirror symmetry, Annales Henri Poincaré, 13, 1873, (2012)
[51] Mironov, A.; Morozov, A.; Morozov, A., Character expansion for HOMFLY polynomials. II. fundamental representation. up to five strands in braid, JHEP, 03, 034, (2012)
[52] Gorsky, E., Combinatorial computation of the motivic Poincaré series, Journal of Singularities, 3, 48, (2011)
[53] Mironov, A.; Morozov, A.; Natanzon, S., Complete set of cut-and-join operators in Hurwitz-Kontsevich theory, Theor. Math. Phys., 166, 1, (2011)
[54] Mironov, A.; Morozov, A.; Natanzon, S., Algebra of differential operators associated with Young diagrams, J. Geom. Phys., 62, 148, (2012)
[55] Morozov, A.; Shakirov, S., Generation of matrix models by W-operators, JHEP, 04, 064, (2009)
[56] Morozov, A.; Shakirov, S., On equivalence of two Hurwitz matrix models, Mod. Phys. Lett., A 24, 2659, (2009)
[57] Borot, G.; Eynard, B.; Mulase, M.; Safnuk, B., A matrix model for simple Hurwitz numbers and topological recursion, J. Geom. Phys., 61, 522, (2011)
[58] Alexandrov, A., Matrix models for random partitions, Nucl. Phys., B 851, 620, (2011)
[59] Awata, H.; Kanno, H., Changing the preferred direction of the refined topological vertex, J. Geom. Phys., 64, 91, (2013)
[60] Awata, H.; Kanno, H., Macdonald operators and homological invariants of the colored Hopf link, J. Phys., A 44, 375201, (2011)
[61] E. Gorsky, \(q\), \(t\)-Catalan numbers and knot homology, arXiv:1003.0916.
[62] Gukov, S.; Sulkowski, P., A-polynomial, B-model and quantization, JHEP, 02, 070, (2012)
[63] N. Carqueville and D. Murfet, Computing Khovanov-Rozansky homology and defect fusion, arXiv:1108.1081 [INSPIRE].
[64] I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE].
[65] S. Shakirov, \(β\)-Deformation and Superpolynomials of (n,m) Torus Knots, arXiv:1111.7035 [INSPIRE].
[66] S. Gukov and M. Stosic, Homological Algebra of Knots and BPS States, arXiv:1112.0030 [INSPIRE].
[67] 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].
[68] Mironov, A.; Morozov, A.; Shakirov, S.; Sleptsov, A., Interplay between Macdonald and Hall-Littlewood expansions of extended torus superpolynomials, JHEP, 05, 070, (2012)
[69] Mironov, A.; Morozov, A.; Shakirov, S., Torus HOMFLY as the Hall-Littlewood polynomials, J. Phys., A 45, 355202, (2012)
[70] H. Fuji, S. Gukov, P. Sulkowski and H. Awata, Volume Conjecture: Refined and Categorified, arXiv:1203.2182 [INSPIRE].
[71] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, A., HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations, JHEP, 07, 131, (2012)
[72] Fuji, H.; Gukov, S.; Sulkowski, P., Super-A-polynomial for knots and BPS states, Nucl. Phys., B 867, 506, (2013)
[73] A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. \(I\). Integrability and difference equations, arXiv:1112.5754 [INSPIRE].
[74] A. Mironov, A. Morozov, A. Sleptsov and A.Smirnov, to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.