×

zbMATH — the first resource for mathematics

Quantum Racah matrices up to level 3 and multicolored link invariants. (English) Zbl 1397.57023
Summary: This paper is a next step in the project of systematic description of colored knot and link invariants started in [A. Mironov and A. Morozov, Nucl. Phys., B 899, 395–413 (2015; Zbl 1331.81264) and A. Mironov et al., J. Phys. A, Math. Theor. 50, No. 8, Article ID 085201, 22 p. (2017; Zbl 1360.81271)]. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels \(R_1 \otimes R_2 \otimes R_3 \longrightarrow Q\) with all possible \(Q\), for \(| R | \leq 3\). The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of a trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
81T45 Topological field theories in quantum mechanics
81P68 Quantum computation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Witten, E., Comm. Math. Phys., 121, 351-399, (1989)
[2] Jones, V. F.R.; Kauffman, L., Invent. Math., Bull. AMS, Ann. of Math., Topology, 26, 395, (1987)
[3] Chern, S.-S.; Simons, J., Ann. of Math., 99, 48-69, (1974)
[4] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B.R.; Millet, K.; Ocneanu, A.; Przytycki, J. H.; Traczyk, K. P., Bull. AMS., Kobe J. Math., 4, 115-139, (1987)
[5] Kauffman, L., Trans. Amer. Math. Soc., 318, 417-471, (1990)
[6] Kaul, R. K.; Govindarajan, T. R.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Sarkar, T.; Zodinmawia; Ramadevi, P., Nuclear Phys. B, Nuclear Phys. B, Nuclear Phys. B, Nuclear Phys. B, Nuclear Phys. B, 870, 205-242, (2013), arXiv:1107.3918arXiv:1209.1346
[7] Ooguri, H.; Vafa, C., Nuclear Phys. B, 577, 419-438, (2000), arXiv:hep-th/9912123
[8] M. Mariño, C. Vafa, hep-th/0108064.; C. Paul, P. Borhade, P. Ramadevi, arXiv:1003.5282; Nucl. Phys. B 841 (2010) 448-462, arXiv:1008.3453.; Wei Luo, Shengmao Zhu, arXiv:1611.06506.; A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, Vivek Kumar Singh, A. Sleptsov, arXiv:1702.06316.; M. Kameyama, S. Nawata, arXiv:1703.05408.
[9] D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov, arXiv:1703.00431.
[10] Mironov, A.; Morozov, A., Nuclear Phys. B, 899, 395-413, (2015), arXiv:1506.00339
[11] Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A., J. Phys. A, 50, (2017), arXiv:1601.04199
[12] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., Phys. Lett. B, 760, 45-58, (2016), arXiv:1605.04881
[13] Nawata, S.; Ramadevi, P.; Zodinmawia, Lett. Math. Phys., 103, 1389-1398, (2013), arXiv:1302.5143
[14] E. Guadagnini, M. Martellini, M. Mintchev, Clausthal 1989, Proceedings, Quantum groups, 307-317; Phys. Lett. B 235 (1990) 275. · Zbl 0768.57003
[15] Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 1203, 034, (2012), arXiv:1112.2654
[16] A. Anokhina, arXiv:1412.8444.; Saswati Dhara, A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, Vivek Kumar Singh, A. Sleptsov, arXiv:1711.10952.
[17] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., J. Mod. Phys. A, 30, (2015), arXiv:1508.02870
[18] Sh. Shakirov, A. Sleptsov, arXiv:1611.03797.
[19] A. Mironov, A. Morozov, arXiv:1610.03043.
[20] Mironov, A.; Morozov, A., Phys. Lett. B, 755, 47-57, (2016), arXiv:1511.09077
[21] C. Bai, J. Jiang, J. Liang, A. Mironov, A. Morozov, An. Morozov, A. Sleptsov, arXiv:1709.09228.
[22] Conway, J. H.; Caudron, A.; Bonahon, F.; Siebenmann, L. C., New geometric splittings of classical knots and the classification and symmetries of arborescent knots, (Leech, John, Computational Problems in Abstract Algebra, Proc. Conf. Oxford, vol. 1967, (1970), Pergamon Press Oxford-New York), Publ. Math. Orsay, 329-358, (2010), University of Paris XI Orsay, http://www-bcf.usc.edu/ fbonahon/Research/Preprints/BonSieb.pdf
[23] Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Nawata, S.; Ramadevi, P.; Zodinmawia; Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A.; Mironov, A.; Morozov, A.; Sleptsov, A.; Nawata, S.; Ramadevi, P.; Singh, Vivek Kumar, Modern Phys. Lett. A, J. Knot Theory and Its Ramifications, Phys. Lett. B, J. High Energy Phys., J. Knot Theor. Ramifications, 26, 069-74, (2017), arXiv:1412.2616. Zodinmawia’s Ph.D. thesis, 2014
[24] Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, V. K., J. High Energy Phys., 1507, 109, (2015), arXiv:1504.00371
[25] Mironov, A.; Morozov, A.; Sleptsov, A., J. High Energy Phys., 07, 069, (2015), arXiv:1412.8432
[26] Dunin-Barkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A.; Mironov, A.; Morozov, A.; Morozov, An.; Arthamonov, S.; Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., AIP Conf. Proc., J. High Energy Phys., 04, 156-155, (2014), arXiv:1309.7984
[27] http://knotebook.org.
[28] J. Gu, H. Jockers, arXiv:1407.5643.
[29] I. Tuba, H. Wenzl, math/9912013.
[30] Anokhina, A.; Morozov, An., Teor. Mat. Fiz., 178, 3-68, (2014), arXiv:1307.2216
[31] http://katlas.org/wiki/The_Thistlethwaite_Link_Table.
[32] Nawata, S.; Ramadevi, P.; Zodinmawia; Sun, X., J. High Energy Phys., 1211, 157, (2012), arXiv:1209.1409
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.