×

zbMATH — the first resource for mathematics

On universal knot polynomials. (English) Zbl 1388.81174
Summary: We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at \(SL\) and \(SO/Sp\) lines on Vogel’s plane, respectively and give their exceptional group’s counterparts on exceptional line. We demonstrate that \([m,n]=[n,m]\) topological invariance, when applicable, take place on the entire Vogel’s plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

MSC:
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] P. Vogel, The universal Lie algebra, preprint (1999) http://webusers.imj-prg.fr/∼pierre.vogel/.
[2] Deligne, P., La série exceptionnelle de groupes de Lie, C. R. Acad. Sci., 322, 321, (1996) · Zbl 0910.22008
[3] Deligne, P.; Man, R., La série exceptionnelle de groupes de Lie II, C. R. Acad. Sci., 323, 577, (1996) · Zbl 0910.22009
[4] Cohen, A.; Man, R., computational evidence for Delignes conjecture regarding exceptional Lie groups, C. R. Acad. Sci., 322, 427, (1996) · Zbl 0849.22017
[5] J.M. Landsberg and L. Manivel, Series of Lie Groups, Michigan Math. J.52 (2004) 453 [math/0203241]. · Zbl 1165.17302
[6] J.M. Landsberg and L. Manivel, Triality, Exceptional Lie Algebras and Deligne Dimension Formulas, Adv. Math.171 (2002) 59 [math/0107032]. · Zbl 1035.17016
[7] Vogel, P., Algebraic structures on modules of diagrams, J. Pure Appl. Algebra, 215, 1292, (2011) · Zbl 1221.57015
[8] J.M. Landsberg and L. Manivel, A universal dimension formula for complex simple Lie algebras, Adv. Math.201 (2006) 379 [math/0401296]. · Zbl 1151.17003
[9] Mkrtchyan, RL; Veselov, AP, Universality in Chern-Simons theory, JHEP, 08, 153, (2012) · Zbl 1397.81326
[10] Katz, SH; Klemm, A.; Vafa, C., Geometric engineering of quantum field theories, Nucl. Phys., B 497, 173, (1997) · Zbl 0935.81058
[11] M. Mariño, Spectral Theory and Mirror Symmetry, arXiv:1506.07757 [INSPIRE].
[12] Gu, J.; Klemm, A.; Mariño, M.; Reuter, J., Exact solutions to quantum spectral curves by topological string theory, JHEP, 10, 025, (2015) · Zbl 1388.81411
[13] Kostant, B., on finite subgroups of SU(2), simple Lie algebras, and the mckay correspondence, Proc. Nat. Acad. Sci., 81, 5275, (1984) · Zbl 0551.22004
[14] Langlands, RP, problems in the theory of automorphic forms to salomon Bochner in gratitude, Springer-verlag, Lect. Notes Math., 170, 18, (1970) · Zbl 0225.14022
[15] Langlands, RP, where stands functoriality today? in representation theory and automorphic forms, Proc. Symp. Pure Math., 61, 457, (1997) · Zbl 0901.11032
[16] A. Beilinson and V. Drinfeld, Quantization of Hitchins integrable system and Hecke eigensheaves, http://www.math.uchicago.edu/∼mitya/langlands/QuantizationHitchin.pdf.
[17] Laumon, G., Correspondance de Langlands géométrique pour LES corps de fonctions, Duke. Math. J., 54, 309, (1987) · Zbl 0662.12013
[18] D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Annals Math.160 (2004) 617 [math/0204081] [INSPIRE]. · Zbl 1129.11050
[19] R. Bezrukavnikov and A. Braverman, Geometric Langlands correspondence for D-modules in prime characteristic: The GL(\(n\)) case, math/0602255 [INSPIRE]. · Zbl 1206.14030
[20] E. Frenkel, Lectures on the Langlands program and conformal field theory, hep-th/0512172 [INSPIRE].
[21] Kapustin, A.; Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys., 1, 1, (2007) · Zbl 1128.22013
[22] Mkrtchyan, RL, Nonperturbative universal Chern-Simons theory, JHEP, 09, 054, (2013) · Zbl 1342.81523
[23] Chern, S-S; Simons, J., Characteristic forms and geometric invariants, Annals Math., 99, 48, (1974) · Zbl 0283.53036
[24] Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121, 351, (1989) · Zbl 0667.57005
[25] Alexander, JW, Topological invariants of knots and links, Trans. Am. Math. Soc., 30, 275, (1928) · JFM 54.0603.03
[26] J.H. Conway, An Enumeration of Knots and Links, and Some of Their Algebraic Properties, in Computational Problems in Abstract Algebra, J. Leech ed., Proc. Conf. Oxford, 1967, Pergamon Press, Oxford-New York (1970), pg. 329-358.
[27] Jones, VFR, Index for subfactors, Invent. Math., 72, 1, (1983) · Zbl 0508.46040
[28] Jones, VFR, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc., 12, 103, (1985) · Zbl 0564.57006
[29] Jones, VFR, Hecke algebra representations of braid groups and link polynomials, Annals Math., 126, 335, (1987) · Zbl 0631.57005
[30] L. Kauffman, State models and the Jones polynomial, Topology26 (1987) 395. x1 · Zbl 0622.57004
[31] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, WBR; Millet, K.; Ocneanu, A., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12, 239, (1985) · Zbl 0572.57002
[32] Przytycki, JH; Traczyk, KP, Invariants of Conway type, Kobe J. Math., 4, 115, (1987) · Zbl 0655.57002
[33] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press (2012), arXiv:1103.5628 [ISBN: 978-1-107-02083-2]. · Zbl 1245.57003
[34] Álvarez-Gaumé, L.; Gomez, C.; Sierra, G., Duality and quantum groups, Nucl. Phys., B 330, 347, (1990) · Zbl 0764.17021
[35] Labastida, JMF; Ramallo, AV, Operator formalism for Chern-Simons theories, Phys. Lett., B 227, 92, (1989)
[36] Labastida, JMF; Ramallo, AV, Chern-Simons and conformal field theories, Nucl. Phys. Proc. Suppl., 16, 594, (1990) · Zbl 0957.81645
[37] Álvarez-Gaumé, L.; Gomez, C.; Sierra, G., Quantum group interpretation of some conformal field theories, Phys. Lett., B 220, 142, (1989) · Zbl 0689.17009
[38] S. Axelrod and I.M. Singer, Chern-Simons perturbation theory, Proc. XXth DGM Conference, New York (1991), S. Catto and A. Rocha eds., World Scientific (1992), pg. 3-45. · Zbl 0813.53051
[39] S. Axelrod and I.M. Singer, Chern-Simons perturbation theory. II, J. Diff. Geom.39 (1994) 173 [hep-th/9304087] [INSPIRE]. · Zbl 0889.53053
[40] Bar-Natan, D., Perturbative Chern-Simons theory, J. Knot Theor., 04, 503, (1995) · Zbl 0861.57009
[41] Bar-Natan, D., On the Vassiliev knot invariants, Topology, 34, 423, (1995) · Zbl 0898.57001
[42] Rosso, M.; Jones, VFR, On the invariants of torus knots derived from quantum groups, J. Knot Theor., 2, 97, (1993) · Zbl 0787.57006
[43] X.-S. Lin and H. Zheng, On the Hecke algebras and the colored HOMFLY polynomial, Trans. Am. Math. Soc.362 (2010) 1 [math/0601267]. · Zbl 1193.57006
[44] Tierz, M., Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett., A 19, 1365, (2004) · Zbl 1076.81544
[45] Brini, A.; Eynard, B.; Mariño, M., Torus knots and mirror symmetry, Annales Henri Poincaré, 13, 1873, (2012) · Zbl 1256.81086
[46] A. Alexandrov, A. Mironov, A. Morozov and An. Morozov, Towards matrix model representation of HOMFLY polynomials, JETP Lett.100 (2014) 271 [arXiv:1407.3754] [INSPIRE].
[47] Dunin-Barkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., Superpolynomials for toric knots from evolution induced by cut-and-join operators, JHEP, 03, 021, (2013) · Zbl 1342.57004
[48] Mironov, A.; Morozov, A.; Morozov, A., Evolution method and “differential hierarchy” of colored knot polynomials, AIP Conf. Proc., 1562, 123, (2013) · Zbl 1291.81260
[49] H. Itoyama, A. Mironov, A. Morozov and An. Morozov, HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations, JHEP07 (2012) 131 [arXiv:1203.5978] [INSPIRE]. · Zbl 1397.57012
[50] Liu, K.; Peng, P., Proof of the labastida-mariño-ooguri-Vafa conjecture, J. Diff. Geom., 85, 479, (2010) · Zbl 1217.81129
[51] Zhu, S., Colored HOMFLY polynomials via skein theory, JHEP, 10, 229, (2013)
[52] N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for knot homologies, math/0505662 [INSPIRE]. · Zbl 1118.57012
[53] Arthamonov, S.; Mironov, A.; Morozov, A., Differential hierarchy and additional grading of knot polynomials, Theor. Math. Phys., 179, 509, (2014) · Zbl 1333.57008
[54] Ya. Kononov and A. Morozov, On the defect and stability of differential expansion, JETP Lett.101 (2015) 831 [arXiv:1504.07146] [INSPIRE]. · Zbl 1388.81337
[55] Mariño, M., String theory and the kauffman polynomial, Commun. Math. Phys., 298, 613, (2010) · Zbl 1207.81129
[56] Guadagnini, E.; Martellini, M.; Mintchev, M., Chern-Simons holonomies and the appearance of quantum groups, Phys. Lett., B 235, 275, (1990) · Zbl 0722.57003
[57] Reshetikhin, NY; Turaev, VG, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., 127, 1, (1990) · Zbl 0768.57003
[58] Kaul, RK; Govindarajan, TR, Three-dimensional Chern-Simons theory as a theory of knots and links, Nucl. Phys., B 380, 293, (1992) · Zbl 0938.81553
[59] 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]. · Zbl 0941.57500
[60] Ramadevi, P.; Govindarajan, TR; Kaul, RK, Knot invariants from rational conformal field theories, Nucl. Phys., B 422, 291, (1994) · Zbl 0990.81694
[61] Ramadevi, P.; Sarkar, T., On link invariants and topological string amplitudes, Nucl. Phys., B 600, 487, (2001) · Zbl 1097.81742
[62] Zodinmawia and P. Ramadevi, SU(\(N\) ) quantum Racah coefficients & non-torus links, Nucl. Phys.B 870 (2013) 205 [arXiv:1107.3918] [INSPIRE]. · Zbl 1262.81168
[63] Zodinmawia and P. Ramadevi, Reformulated invariants for non-torus knots and links, arXiv:1209.1346 [INSPIRE]. · Zbl 1262.81168
[64] A. Mironov, A. Morozov and An. Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, in Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, World Scientific (2013), pg. 101-118, arXiv:1112.5754.
[65] A. Mironov, A. Morozov and An. Morozov, Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid, JHEP03 (2012) 034 [arXiv:1112.2654] [INSPIRE]. · Zbl 1309.81114
[66] H. Itoyama, A. Mironov, A. Morozov and An. 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]. · Zbl 1260.81134
[67] A. Anokhina, A. Mironov, A. Morozov and An. 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]. · Zbl 1262.81073
[68] H. Itoyama, A. Mironov, A. Morozov and An. 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]. · Zbl 1259.81082
[69] A. Anokhina, A. Mironov, A. Morozov and An. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Adv. High Energy Phys.2013 (2013) 931830 [arXiv:1304.1486] [INSPIRE].
[70] A. Anokhina and An. Morozov, Cabling procedure for the colored HOMFLY polynomials, Teor. Mat. Fiz.178 (2014) 3 [arXiv:1307.2216] [INSPIRE]. · Zbl 1318.81055
[71] S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theor.22 (2013) 1350078 [arXiv:1302.5144] [INSPIRE]. · Zbl 1296.57015
[72] Zodinmawia, Knot polynomials from SU(\(N\) ) Chern-Simons theory, superpolynomials and super-A-polynomials, Ph.D. Thesis (2014).
[73] Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A., Colored knot polynomials for arbitrary pretzel knots and links, Phys. Lett., B 743, 71, (2015) · Zbl 1343.57007
[74] Mironov, A.; Morozov, A.; Sleptsov, A., Colored HOMFLY polynomials for the pretzel knots and links, JHEP, 07, 069, (2015) · Zbl 1388.57012
[75] S. Nawata, P. Ramadevi and V.K. Singh, Colored HOMFLY polynomials can distinguish mutant knots, arXiv:1504.00364 [INSPIRE]. · Zbl 1405.57017
[76] A. Mironov, A. Morozov, An. Morozov, P. Ramadevi and V.K. Singh, Colored HOMFLY polynomials of knots presented as double fat diagrams, JHEP07 (2015) 109 [arXiv:1504.00371] [INSPIRE]. · Zbl 1388.57010
[77] Mironov, A.; Morozov, A., Towards effective topological field theory for knots, Nucl. Phys., B 899, 395, (2015) · Zbl 1331.81264
[78] A. Mironov, A. Morozov, An. Morozov and A. Sleptsov, Colored knot polynomials: HOMFLY in representation [2\(,\) 1], Int. J. Mod. Phys.A 30 (2015) 1550169 [arXiv:1508.02870] [INSPIRE]. · Zbl 1333.81202
[79] Aganagic, M.; Shakirov, S., Knot homology and refined Chern-Simons index, Commun. Math. Phys., 333, 187, (2015) · Zbl 1322.81069
[80] I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE]. · Zbl 1329.57019
[81] Mironov, A.; Morozov, A.; Shakirov, S.; Sleptsov, A., Interplay between Macdonald and Hall-Littlewood expansions of extended torus superpolynomials, JHEP, 05, 070, (2012) · Zbl 1348.81391
[82] Gorsky, E.; Negut, A., Refined knot invariants and Hilbert schemes, J. Math. Pure. Appl., 104, 403, (2015) · Zbl 1349.14012
[83] I. Cherednik and I. Danilenko, DAHA and iterated torus knots, arXiv:1408.4348.
[84] A. Mironov, A. Morozov and An. Morozov, On colored HOMFLY polynomials for twist knots, Mod. Phys. Lett.A 29 (2014) 1450183 [arXiv:1408.3076] [INSPIRE].
[85] Mironov, A.; Morozov, A.; Natanzon, S., Complete set of cut-and-join operators in Hurwitz-Kontsevich theory, Theor. Math. Phys., 166, 1, (2011) · Zbl 1312.81125
[86] Mironov, A.; Morozov, A.; Natanzon, S., Algebra of differential operators associated with Young diagrams, J. Geom. Phys., 62, 148, (2012) · Zbl 1242.22008
[87] Bouchard, V.; Florea, B.; Mariño, M., Counting higher genus curves with crosscaps in Calabi-Yau orientifolds, JHEP, 12, 035, (2004)
[88] Stevan, S., Chern-Simons invariants of torus links, Annales Henri Poincaré, 11, 1201, (2010) · Zbl 1208.81149
[89] D. Bar-Natan, http://katlas.org.
[90] C. Livingston, http://www.indiana.edu/∼knotinfo/.
[91] Mkrtchian, RL, the equivalence of sp(2\(N\) ) and SO(−2\(N\) ) gauge theories, Phys. Lett., B 105, 174, (1981)
[92] P. Cvitanović, Group Theory, Princeton University Press, Princeton, NJ (2004), http://www.nbi.dk/grouptheory.
[93] R. Mkrtchyan, unpublished (2013).
[94] P.Deligne, unpublished (2013).
[95] Okubo, S., Casimir invariants and vector operators in simple Lie algebra, J. Math. Phys., 18, 2382, (1977) · Zbl 0371.17004
[96] Mkrtchyan, RL; Sergeev, AN; Veselov, AP, Casimir eigenvalues for universal Lie algebra, J. Math. Phys., 53, 102106, (2012) · Zbl 1331.17007
[97] Standard representation of symmetric group: S3, http://groupprops.subwiki.org/wiki/Standard representation of symmetric group:S3.
[98] S. Nawata, P. Ramadevi and Zodinmawia, Colored Kauffman Homology and Super-A-polynomials, JHEP01 (2014) 126 [arXiv:1310.2240] [INSPIRE]. · Zbl 1333.81351
[99] R. Hadji and H. Morton, A basis for the full Homfly skein of the annulus, Math. Proc. Camb. Philos. Soc.141 (2006) 81 [math/0408078]. · Zbl 1108.57005
[100] A. Anokhina, A. Mironov, A. Morozov and An. Morozov, Knot polynomials in the first non-symmetric representation, Nucl. Phys.B 882 (2014) 171 [arXiv:1211.6375] [INSPIRE]. · Zbl 1285.81035
[101] B.W. Westbury, Extending and quantising the Vogel plane, arXiv:1510.08307.
[102] A. Mironov and A. Morozov, Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots, arXiv:1511.09077 [INSPIRE]. · Zbl 1367.81090
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.