Alexander Polynomial Invariants of Torus Knots T(n, 3) and Chebyshev Polynomials
DOI:
https://doi.org/10.15407/ujpe56.7.680Keywords:
-Abstract
The explicit formula, which expresses the Alexander polynomials ∆n,3(t) of torus knots T(n, 3) as a sum of the Alexander polynomials ∆k,2(t) of torus knots T(k, 2), is found. Using this result and those from our previous papers, we express the Alexander polynomials ∆n,3(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n, l) with n and l coprime.
References
M.F. Atiyah, The Geometry and Physics of Knots (Cambridge Univ. Press, Cambridge, 1990).
https://doi.org/10.1017/CBO9780511623868
L.H. Kauffman, Knots and Physics (World Scientific, Singapore, 2001).
The Interface of Knots and Physics, AMS Short Course Lecture Notes, Proc. Symp. App. Math. 51, edited by L.H. Kauffman (Amer. Math. Soc., Providence, RI, 1996).
M.D. Frank-Kamenetskii and A.V. Vologodskii, Uspekhi Fiz. Nauk 134, 641 (1981).
https://doi.org/10.3367/UFNr.0134.198108c.0641
L.D. Faddeev, Quantization of Solitons, Princeton preprint IAS-75-QS70 (Institute for Advanced Study, Princeton, 1975).
L. Faddeev and Antti J. Niemi, Nature 387, 58 (1997); arXiv:hep-th/9610193, 1996.
https://doi.org/10.1038/387058a0
R.A. Battye and P.M. Sutcliffe, Phys. Rev. Lett. 81, 4798 (1998)
https://doi.org/10.1103/PhysRevLett.81.4798
Proc. R. Soc. Lon. A 455, 4305 (1999).
https://doi.org/10.1098/rspa.1999.0502
J. Gladikowski and M. Hellmund, Phys. Rev. D 56, 5194 (1997).
https://doi.org/10.1103/PhysRevD.56.5194
J. Hietarinta and P. Salo, Phys. Lett. B 451, 60 (1999)
https://doi.org/10.1016/S0370-2693(99)00054-4
Phys. Rev. D 62, 081701 (2000).
P. Sutcliffe, Proc. Roy. Soc. Lond. A 463, 3001 (2007).
https://doi.org/10.1098/rspa.2007.0038
E. Radu and M.S. Volkov, Phys. Rep. 468, No. 4, 101 (2008).
https://doi.org/10.1016/j.physrep.2008.07.002
A.L. Kholodenko, arXiv:1001.0029v1 [hep-th], 2010; J. Geom. Phys. 33, 23 (2000); arXiv:gr-qc/0010064v1, 2000.
H. Jehle, Phys. Rev. D 6, 441 (1972); Phys. Lett. B 104, 207 (1981).
https://doi.org/10.1103/PhysRevD.6.441
A.M. Gavrilik, J. Phys. A 27, 91 (1994)
https://doi.org/10.1088/0305-4470/27/3/006
Nucl. Phys. B (Proc. Suppl.) 102, 298 (2001).
https://doi.org/10.1016/S0920-5632(01)01570-5
R.J. Finkelstein, Int. J. Mod. Phys. A 22, 4467 (2007).
https://doi.org/10.1142/S0217751X0703707X
E. Witten, Comm. Math. Phys. 121, 351 (1989).
https://doi.org/10.1007/BF01217730
A.M. Gavrilik and A.M. Pavlyuk, Ukr. J. Phys. 55, 129 (2010); arXiv:0912.4674v2 [math-ph], 2009.
P.-V. Koseleff and D. Pecker, arXiv:0712.2408v1 [math.HO], 2007; arXiv:0812.1089v1 [math.GT], 2008.
D. Rolfsen, Knots and Links (Amer. Math. Soc., Providence, RI, 2003).
https://doi.org/10.1090/chel/346
L.C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989).
https://doi.org/10.1088/0305-4470/22/18/004
A.J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989).
https://doi.org/10.1088/0305-4470/22/21/020
I.I. Kachurik, Ukr. J. Math. 50, 1055 (1998).
https://doi.org/10.1007/BF02513092
V. Kac and P. Cheung, Quantum Calculus (Springer, New York, 2002).
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