This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation $R = [2, 1]$ $R = [2, 1]$ . The project involves several steps:

⁽ⁱ⁾	parametrization of big families of knots á la [A. Mironov and A. Morozov, arXiv:1506.00339],
⁽ⁱⁱ⁾	evaluating Racah/mixing matrices for various numbers of strands in various representations á la [A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys.03, 034 (2012), arXiv:1112.2654],
⁽ⁱⁱⁱ⁾	tabulating and collecting the results at http://www.knotebook.org.

In this paper, we discuss only the representation $R = [2, 1]$ $R = [2, 1]$ and construct all necessary ingredients that allow one to evaluate knot/links represented by three-strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family. This family contains over 80% of knots with up to 10 intersections, but does not include mutants.

Keywords:

PACS: 11.15.Yc, 02.10.Kn, 02.20.Uw

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

1. A. Mironov and A. Morozov, arXiv:1506.00339. Google Scholar
2. A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654. Crossref, Web of Science, ADS, Google Scholar
3. http://www.knotebook.org. Google Scholar
4. J. W. Alexander, Trans. Amer. Math. Soc. 30, 275 (1928). Crossref, Google Scholar
5. J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra, Proc. Conf., Oxford, 1967, ed. J. Leech (Pergamon Press, Oxford, 1970), pp. 329–358. Google Scholar
6. V. F. R. Jones, Invent. Math. 72, 1 (1983). Crossref, Web of Science, ADS, Google Scholar
7. V. F. R. Jones, Bull. AMS 12, 103 (1985). Crossref, Web of Science, Google Scholar
8. V. F. R. Jones, Ann. Math. 126, 335 (1987). Crossref, Web of Science, Google Scholar
9. L. Kauffman, Topology 26, 395 (1987). Crossref, Web of Science, Google Scholar
10. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, Bull. AMS 12, 239 (1985). Crossref, Web of Science, Google Scholar
11. J. H. Przytycki and K. P. Traczyk, Kobe J. Math. 4, 115 (1987). Web of Science, Google Scholar
12. S.-S. Chern and J. Simons, Ann. Math. 99, 48 (1974). Crossref, Web of Science, Google Scholar
13. E. Witten, Commun. Math. Phys. 121, 351 (1989). Crossref, Web of Science, ADS, Google Scholar
14. R. K. Kaul and T. R. Govindarajan, Nucl. Phys. B 380, 293 (1992), arXiv:hep-th/9111063. Crossref, Web of Science, ADS, Google Scholar
15. P. Ramadevi, T. R. Govindarajan and R. K. Kaul, Nucl. Phys. B 402, 548 (1993), arXiv:hep-th/9212110. Crossref, Web of Science, ADS, Google Scholar
16. P. Ramadevi, T. R. Govindarajan and R. K. Kaul, Nucl. Phys. B 422, 291 (1994), arXiv:hep-th/9312215. Crossref, Web of Science, ADS, Google Scholar
17. P. Ramadevi and T. Sarkar, Nucl. Phys. B 600, 487 (2001), arXiv:hep-th/0009188. Crossref, Web of Science, ADS, Google Scholar
18. Zodinmawia and P. Ramadevi, arXiv:1107.3918. Google Scholar
19. Zodinmawia and P. Ramadevi, arXiv:1209.1346. Google Scholar
20. D. Galakhov, A. Mironov and A. Morozov, JETP 120, 623 (2015) [ZhETF 147, 623 (2015)], arXiv:1410.8482. Crossref, Web of Science, Google Scholar
21. D. Galakhov, D. Melnikov, A. Mironov and A. Morozov, arXiv:1502.02621. Google Scholar
22. O. Alekseev and F. Novaes, arXiv:1505.06221. Google Scholar
23. M. Tierz, Mod. Phys. Lett. A 19, 1365 (2004), arXiv:hep-th/0212128. Link, Web of Science, ADS, Google Scholar
24. A. Brini, B. Eynard and M. Mariño, Ann. Henri Poincaré 13, 1873 (2012), arXiv:1105.2012. Crossref, Web of Science, ADS, Google Scholar
25. A. Alexandrov, A. Mironov, A. Morozov and An. Morozov, JETP Lett. 100, 271 (2014) [Pis’ma v ZhETF 100, 297 (2014)], arXiv:1407.3754. Crossref, Web of Science, ADS, Google Scholar
26. A. Morozov, Phys. Usp. (UFN) 35, 671 (1992). Crossref, ADS, Google Scholar
27. A. Morozov, Phys. Usp. (UFN) 37, 1 (1994), arXiv:hep-th/9303139. Crossref, Web of Science, ADS, Google Scholar
28. A. Morozov, arXiv:hep-th/9502091. Google Scholar
29. A. Morozov, arXiv:hep-th/0502010. Google Scholar
30. A. Mironov, Int. J. Mod. Phys. A 9, 4355 (1994), arXiv:hep-th/9312212. Link, Web of Science, ADS, Google Scholar
31. A. Mironov, Phys. Part. Nucl. 33, 537 (2002). Web of Science, Google Scholar
32. A. Mironov, A. Morozov and A. Sleptsov, Eur. Phys. J. C 73, 2492 (2013), arXiv:1304.7499. Crossref, Web of Science, ADS, Google Scholar
33. H. Ooguri and C. Vafa, Nucl. Phys. B 577, 419 (2000), arXiv:hep-th/9912123. Crossref, Web of Science, ADS, Google Scholar
34. A. Mironov, A. Morozov and An. Morozov, Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, eds. A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger (World Scientific, 2013), pp. 101–118, arXiv:1112.5754. Google Scholar
35. M. Aganagic and C. Vafa, arXiv:1204.4709. Google Scholar
36. M. Aganagic, T. Ekholm, L. Ng and C. Vafa, arXiv:1304.5778. Google Scholar
37. S. Arthamonov, A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys. 04, 156 (2014), arXiv:1309.7984. Crossref, Web of Science, ADS, Google Scholar
38. R. Gelca, Math. Proc. Camb. Philos. Soc. 133, 311 (2002), arXiv:math/0004158. Crossref, Web of Science, Google Scholar
39. R. Gelca and J. Sain, J. Knot Theory Ramif. 12, 187 (2003), arXiv:math/0201100. Link, Web of Science, Google Scholar
40. S. Gukov, Commun. Math. Phys. 255, 577 (2005), arXiv:hep-th/0306165. Crossref, Web of Science, ADS, Google Scholar
41. S. Garoufalidis, Geom. Topol. Monogr. 7, 291 (2004), arXiv:math/0306230. Crossref, Google Scholar
42. A. Mironov and A. Morozov, AIP Conf. Proc. 1483, 189 (2012), arXiv:1208.2282. Google Scholar
43. M. Mariño, Commun. Math. Phys. 298, 613 (2010), arXiv:0904.1088. Crossref, Web of Science, ADS, Google Scholar
44. L. H. Kauffman, Eur. J. Comb. 20, 663 (1999), arXiv:math/9811028. Crossref, Web of Science, Google Scholar
45. R. Fenn, D. P. Ilyutko, L. H. Kauffman and V. O. Manturov, arXiv:1409.2823. Google Scholar
46. A. Morozov, An. Morozov and Ant. Morozov, Phys. Lett. B 737, 48 (2014), arXiv:1407.6319. Crossref, Web of Science, ADS, Google Scholar
47. L. Bishler, A. Morozov, An. Morozov and Ant. Morozov, Int. J. Mod. Phys. A 30, 1550074 (2015), arXiv:1411.2569. Link, Web of Science, ADS, Google Scholar
48. A. Morozov, An. Morozov and A. Popolitov, arXiv:1506.07516. Google Scholar
49. N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990). Crossref, Web of Science, Google Scholar
50. E. Guadagnini, M. Martellini and M. Mintchev, Proc. Quantum Groups, Clausthal, 1989, pp. 307–317. Google Scholar
51. E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B 235, 275 (1990). Crossref, Web of Science, ADS, Google Scholar
52. V. G. Turaev and O. Y. Viro, Topology 31, 865 (1992). Crossref, Web of Science, Google Scholar
53. A. Morozov and A. Smirnov, Nucl. Phys. B 835, 284 (2010), arXiv:1001.2003. Crossref, Web of Science, ADS, Google Scholar
54. A. Smirnov, Proc. of Int. School of Subnuclear Physics, Erice, Italy, 2009, arXiv:0910.5011 [hep-th]. Google Scholar
55. H. Itoyama, A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys. 2012, 131 (2012), arXiv:1203.5978. Crossref, Web of Science, Google Scholar
56. H. Itoyama, A. Mironov, A. Morozov and An. Morozov, Int. J. Mod. Phys. A 27, 1250099 (2012), arXiv:1204.4785. Link, Web of Science, ADS, Google Scholar
57. H. Itoyama, A. Mironov, A. Morozov and An. Morozov, Int. J. Mod. Phys. A 28, 1340009 (2013), arXiv:1209.6304. Link, Web of Science, ADS, Google Scholar
58. A. Anokhina, A. Mironov, A. Morozov and An. Morozov, Nucl. Phys. B 868, 271 (2013), arXiv:1207.0279. Crossref, Web of Science, ADS, Google Scholar
59. A. Anokhina, A. Mironov, A. Morozov and An. Morozov, Adv. High Energy Phys. 2013, 931830 (2013), arXiv:1304.1486. Crossref, Web of Science, Google Scholar
60. A. Anokhina and A. Morozov, Teor. Mat. Fiz. 178, 3 (2014) [Theor. Math. Phys. 178, 1 (2014)], arXiv:1307.2216. Crossref, Web of Science, Google Scholar
61. S. Nawata, P. Ramadevi and V. K. Singh, arXiv:1504.00364. Google Scholar
62. A. Mironov, A. Morozov, An. Morozov, P. Ramadevi and V. K. Singh, J. High Energy Phys. 1507, 109 (2015), arXiv:1504.00371. Crossref, Web of Science, ADS, Google Scholar
63. M. Khovanov, Duke Math. J. 101, 359 (2000), arXiv:math/9908171. Crossref, Web of Science, Google Scholar
64. M. Khovanov, Exp. Math. 12, 365 (2003), arXiv:math/0201306. Crossref, Web of Science, Google Scholar
65. M. Khovanov, J. Knot Theory Ramif. 14, 111 (2005), arXiv:math/0302060. Link, Web of Science, Google Scholar
66. M. Khovanov, Algebr. Geom. Topol. 4, 1045 (2004), arXiv:math/0304375. Crossref, Google Scholar
67. M. Khovanov, Int. J. Math. 18, 869 (2007), arXiv:math/0510265. Link, Web of Science, Google Scholar
68. M. Khovanov, arXiv:math/0605339. Google Scholar
69. M. Khovanov, arXiv:1008.5084. Google Scholar
70. D. Bar-Natan, Algebr. Geom. Topol. 2, 337 (2002), arXiv:math/0201043. Crossref, Google Scholar
71. D. Bar-Natan, Geom. Topol. 9, 1443 (2005), arXiv:math/0410495. Crossref, Web of Science, Google Scholar
72. D. Bar-Natan, J. Knot Theory Ramif. 16, 243 (2007), arXiv:math/0606318. Link, Web of Science, Google Scholar
73. M. Khovanov and L. Rozansky, Fund. Math. 199, 191 (2008), arXiv:math/0401268. Crossref, Web of Science, Google Scholar
74. M. Khovanov and L. Rozansky, Geom. Topol. 12, 13871425 (2008), arXiv:math/0505056. Crossref, Web of Science, Google Scholar
75. M. Khovanov and L. Rozansky, arXiv:math/0701333. Google Scholar
76. N. Carqueville and D. Murfet, arXiv:1108.1081. Google Scholar
77. V. Dolotin and A. Morozov, J. High Energy Phys. 1301, 065 (2013), arXiv:1208.4994. Crossref, Web of Science, ADS, Google Scholar
78. V. Dolotin and A. Morozov, J. Phys.: Conf. Ser. 411, 012013 (2013), arXiv:1209.5109. Google Scholar
79. V. Dolotin and A. Morozov, Nucl. Phys. B 878, 12 (2014), arXiv:1308.5759. Crossref, Web of Science, ADS, Google Scholar
80. A. Anokhina and A. Morozov, J. High Energy Phys. 07, 063 (2014), arXiv:1403.8087. Crossref, Web of Science, ADS, Google Scholar
81. H. Morton and S. Lukac, J. Knot Theory Ramif. 12, 395 (2003), arXiv:math.GT/0108011. Link, Web of Science, Google Scholar
82. J. Gu and H. Jockers, arXiv:1407.5643. Google Scholar
83. A. Mironov, A. Morozov and A. Sleptsov, J. High Energy Phys. 07, 069 (2015), arXiv:1412.8432. Crossref, Web of Science, ADS, Google Scholar
84. M. Rosso and V. F. R. Jones, J. Knot Theory Ramif. 2, 97 (1993). Link, Google Scholar
85. X.-S. Lin and H. Zheng, Trans. Amer. Math. Soc. 362, 1 (2010), arXiv:math/0601267. Crossref, Web of Science, Google Scholar
86. A. Mironov, A. Morozov and An. Morozov, AIP Conf. Proc. 1562, 123 (2013), arXiv:1306.3197. Google Scholar
87. A. Mironov, A. Morozov and An. Morozov, Mod. Phys. Lett. A 29, 1450183 (2014), arXiv:1408.3076. Link, Web of Science, ADS, Google Scholar
88. S. Arthamonov, A. Mironov and A. Morozov, Theor. Math. Phys. 179, 509 (2014) [Teor. Mat. Fiz. 179, 147 (2014)], arXiv:1306.5682. Crossref, Web of Science, Google Scholar
89. D. Bar-Natan, http://katlas.org. Google Scholar
90. Ch. Livingston, http://http://www.indiana.edu/~knotinfo/. Google Scholar
91. D. Bar-Natan and J. Greene, http://www.math.toronto.edu/~drorbn/Students/GreenJ/. Google Scholar
92. P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, J. High Energy Phys. 03, 021 (2013), arXiv:1106.4305. Crossref, Web of Science, ADS, Google Scholar
93. D. Galakhov, D. Melnikov, A. Mironov, A. Morozov and A. Sleptsov, Phys. Lett. B 743, 71 (2015), arXiv:1412.2616. Crossref, Web of Science, ADS, Google Scholar
94. P. Ramadevi, T. R. Govindarajan and R. K. Kaul, Mod. Phys. Lett. A 9, 3205 (1994), arXiv:hep-th/9401095. Link, Web of Science, ADS, Google Scholar
95. S. Nawata, P. Ramadevi and Zodinmawia, J. Knot Theory Ramif. 22, 13 (2013), arXiv:1302.5144. Link, Web of Science, Google Scholar
96. Zodinmawia, Ph.D. thesis (2014). Google Scholar
97. I. Tuba and H. Wenzl, arXiv:math/9912013. Google Scholar
98. N. M. Dunfield, S. Gukov and J. Rasmussen, Exp. Math. 15, 129 (2006), arXiv:math/0505662. Crossref, Web of Science, Google Scholar
99. A. Anokhina, A. Mironov, A. Morozov and An. Morozov, Nucl. Phys. B 882C, 171 (2014), arXiv:1211.6375. Crossref, Web of Science, ADS, Google Scholar
100. Ya. Kononov and A. Morozov, Pis’ma v ZhETF 101, 931 (2015), arXiv:1504.07146. Google Scholar
101. Ya. Kononov and A. Morozov, Phys. Lett. B 747, 500 (2015), arXiv:1505.06170. Crossref, Web of Science, ADS, Google Scholar