×

zbMATH — the first resource for mathematics

Differential expansion and rectangular HOMFLY for the figure eight knot. (English) Zbl 1346.81079
Summary: Differential expansion (DE) for a Wilson loop average in representation \(R\) is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of 3\(d\) Chern-Simons theory. Especially simple is the relation between the DE for the trefoil \(3_{1}\) and for the figure eight knot \(4_{1}\). Since arbitrary colored HOMFLY for the trefoil are known from the Rosso-Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfill this program for all rectangular representation \(R = [r^s]\), i.e. make a plausible conjecture for the rectangularly colored HOMFLY of the figure eight knot, which generalizes the old result for totally symmetric and antisymmetric representations.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81T45 Topological field theories in quantum mechanics
30C35 General theory of conformal mappings
32S55 Milnor fibration; relations with knot theory
Keywords:
HOMFLY
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Chern, S.-S.; Simons, J.; Witten, E., Ann. Math., Commun. Math. Phys., 121, 351-399, (1989)
[2] Kononov, Ya.; Morozov, A.
[3] Dolotin, V.; Morozov, A.; Dolotin, V.; Morozov, A.; Dolotin, V.; Morozov, A., Int. J. Mod. Phys. A, 23, 3613-3684, (2008) · Zbl 1284.81150
[4] Alexander, J. W.; Jones, V. F.R.; Jones, V. F.R.; Jones, V. F.R.; Kauffman, L.; Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B.R.; Millet, K.; Ocneanu, A.; Przytycki, J. H.; Traczyk, K. P., Trans. Am. Math. Soc., Invent. Math., Bull. Am. Math. Soc., Ann. Math., Topology, Bull. Am. Math. Soc., Kobe J. Math., 4, 2, 115-139, (1987)
[5] Conway, J. H., Algebraic properties, (Leech, John, Computational Problems in Abstract Algebra, Proc. Conf. Oxford, 1967, (1970), Pergamon Press Oxford-New York), 329-358
[6] Atiyah, M., The geometry and physics of knots, (1990), CUP · Zbl 0729.57002
[7] Bar-Natan, D.; Morrison, S.
[9] Alexandrov, A.; Mironov, A.; Morozov, A.; Alexandrov, A.; Mironov, A.; Morozov, A.; Eynard, B.; Orantin, N.; Orantin, N., Physica D, J. High Energy Phys., Commun. Number Theory Phys., 1, 347-452, (2007)
[10] Dunfield, N. M.; Gukov, S.; Rasmussen, J., Exp. Math., 15, 129-159, (2006)
[11] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 2012, (2012)
[12] Reshetikhin, N. Yu.; Turaev, V. G.; Guadagnini, E.; Martellini, M.; Mintchev, M.; Guadagnini, E.; Martellini, M.; Mintchev, M.; Turaev, V. G.; Viro, O. Y.; Morozov, A.; Smirnov, A.; Smirnov, A., Proc. of international school of subnuclar phys., (Clausthal Procs., (1989)), Phys. Lett. B, Topology, Nucl. Phys. B, 835, 284-313, (2009), Erice, Italy
[13] Govindarajan, T. R.; Kaul, R. K.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Sarkar, T.; Ramadevi, P.; Zodinmawia; Ramadevi, P.; Zodinmawia, Nucl. Phys. B, Nucl. Phys. B, Nucl. Phys. B, Nucl. Phys. B, Mod. Phys. Lett. A, Nucl. Phys. B, 600, 487-511, (2001)
[14] Rozansky, L.; Morozov, A., Theor. Math. Phys., 187, 447-454, (2016)
[15] Khovanov, M.; Khovanov, M.; Khovanov, M.; Khovanov, M.; Khovanov, M.; Khovanov, M.; Khovanov, M.; Bar-Natan, D.; Bar-Natan, D.; Bar-Natan, D.; Khovanov, M.; Rozansky, L.; Khovanov, M.; Rozansky, L.; Khovanov, M.; Rozansky, L.; Carqueville, N.; Murfet, D.; Dolotin, V.; Morozov, A.; Dolotin, V.; Morozov, A.; Dolotin, V.; Morozov, A.; Anokhina, A.; Morozov, A.; Nawata, S.; Oblomkov, A., Duke Math. J., Exp. Math., J. Knot Theory Ramif., Algebraic Geom. Topol., Int. J. Math., Algebraic Geom. Topol., Geom. Topol., J. Knot Theory Ramif., Fundam. Math., Geom. Topol., J. High Energy Phys., J. Phys., Nucl. Phys. B, J. High Energy Phys., 07, 3, 12-81, (2014)
[16] Gopakumar, R.; Vafa, C.; Ooguri, H.; Vafa, C., Adv. Theor. Math. Phys., Nucl. Phys. B, 577, 419-438, (2000)
[17] Liu, K.; Peng, P.
[18] Aganagic, M.; Shakirov, Sh.; Aganagic, M.; Shakirov, Sh.; Aganagic, M.; Shakirov, Sh.; Mironov, A.; Morozov, A.; Shakirov, Sh.; Sleptsov, A.; Gorsky, E.; Negut, A.; Cherednik, I.; Danilenko, I.; Arthamonov, S.; Shakirov, Sh., J. High Energy Phys., 2012, (2012)
[19] Dunin-Barkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., J. High Energy Phys., 03, (2013)
[20] Cherednik, I.
[21] Gukov, S.; Stosic, M.
[22] Mironov, A.; Morozov, A.; Morozov, An.; Mironov, A.; Morozov, A.; Morozov, An.; Anokhina, A.; Mironov, A.; Morozov, A.; Morozov, An.; Anokhina, A., (Rebhan, A.; Katzarkov, L.; Knapp, J.; Rashkov, R.; Scheidegger, E., Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, (2013), World Scietific Publishins Co. Pte. Ltd.), J. High Energy Phys., Nucl. Phys. B, 868, 271-313, (2013) · Zbl 1262.81073
[23] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An.; Nawata, S.; Ramadevi, P.; Zodinmawia; Sun, X.; Fuji, H.; Gukov, S.; Stosic, M.; Sulkowski, P., Int. J. Mod. Phys. A, J. High Energy Phys., 1211, 1250099, (2012)
[24] Liu, Kefeng; Peng, Pan; Zhu, Shengmao; Morozov, Anton; Morozov, Anton, J. High Energy Phys., JETP Lett., 12, (2013)
[25] Mironov, A.; Morozov, A., AIP Conf. Proc., 1483, 189-211, (2012)
[26] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An.; Tuba, I.; Wenzl, H., Int. J. Mod. Phys. A, 28, 1340009, (2013), see also
[27] Anokhina, A.; Mironov, A.; Morozov, A.; Morozov, An., Nucl. Phys. B, 882C, 171-194, (2014)
[28] Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Nawata, S.; Ramadevi, P.; Zodinmawia; Nawata, S.; Ramadevi, P.; Zodinmawia; Nawata, S.; Ramadevi, P.; Zodinmawia; Nawata, S.; Ramadevi, P.; Zodinmawia, Mod. Phys. Lett. A, J. High Energy Phys., J. High Energy Phys., 1401, 3205-3218, (2014), Zodinmawia’s PhD thesis
[29] Mironov, A.; Morozov, A.; Sleptsov, A.; Mironov, A.; Morozov, A.; Sleptsov, A.; Sleptsov, A., Theor. Math. Phys., Eur. Phys. J. C, Int. J. Mod. Phys. A, 29, 1430063-221, (2014)
[30] Anokhina, A.; Mironov, A.; Morozov, A.; Morozov, An., Adv. High Energy Phys., 2013, 931830, (2013)
[31] Gorsky, E.; Gukov, S.; Stosic, M.
[32] Mironov, A.; Morozov, A.; Morozov, An., AIP Conf. Proc., 1562, 123, (2013)
[33] Arthamonov, S.; Mironov, A.; Morozov, A., Theor. Math. Phys., 179, 509-542, (2014)
[34] Anokhina, A.; Morozov, An., Theor. Math. Phys., 178, 1-58, (2014)
[35] Arthamonov, S.; Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 04, (2014)
[36] Nawata, S.; Ramadevi, P.; Zodinmawia; Gukov, S.; Nawata, S.; Saberi, I.; Stosic, M.; Sulkowski, P., J. High Energy Phys., 1401, (2014)
[37] Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., Nucl. Phys. B, 889, 757-777, (2014)
[38] Witten, E.
[39] Gu, J.; Jockers, H.
[40] Mironov, A.; Morozov, A.; Morozov, An., Mod. Phys. Lett. A, 29, 1450183, (2014)
[41] Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A.; Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A., Phys. Lett. B, Nucl. Phys. B, 899, 194-228, (2015)
[42] Mironov, A.; Morozov, A.; Sleptsov, A., J. High Energy Phys., 07, (2015)
[43] Nawata, S.; Ramadevi, P.; Singh, Vivek Kumar
[44] Kononov, Ya.; Morozov, A., Pis’ma v ZhETF, 101, 931-934, (2015)
[45] Kononov, Ya.; Morozov, A., Phys. Lett. B, 747, 500-510, (2015)
[46] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., J. Mod. Phys. A, 30, 1550169, (2015)
[47] Galakhov, D.; Mironov, A.; Morozov, A.
[48] Mironov, A.; Mkrtchyan, R.; Morozov, A., J. High Energy Phys., 02, (2016)
[49] Garoufalidis, S.; Lauda, A. D.; Lê, T. T.Q.
[50] Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, Vivek Kumar, J. High Energy Phys., 1507, (2015)
[51] Mironov, A.; Morozov, A., Nucl. Phys. B, 899, 395-413, (2015)
[52] Mironov, A.; Morozov, A., Phys. Lett. B, 755, 47-57, (2016)
[53] Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.
[54] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.
[55] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., Pis’ma Zh. Eksp. Teor. Fiz., 104, 52-57, (2016)
[56] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., Phys. Lett. B, 760, 45-58, (2016)
[57] Rosso, M.; Jones, V. F.R.; Lin, X.-S.; Zheng, H.; Tierz, M.; Brini, A.; Eynard, B.; Marino, M.; Alexandrov, A.; Mironov, A.; Morozov, A.; Morozov, An., J. Knot Theory Ramif., Trans. Am. Math. Soc., Mod. Phys. Lett. A, JETP Lett., 100, 271-278, (2014)
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.