Alexander Polynomial Invariants of Torus Knots T(n, 3) and Chebyshev Polynomials
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.
M.F. Atiyah, The Geometry and Physics of Knots (Cambridge Univ. Press, Cambridge, 1990).
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).
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.
R.A. Battye and P.M. Sutcliffe, Phys. Rev. Lett. 81, 4798 (1998)
Proc. R. Soc. Lon. A 455, 4305 (1999).
J. Gladikowski and M. Hellmund, Phys. Rev. D 56, 5194 (1997).
J. Hietarinta and P. Salo, Phys. Lett. B 451, 60 (1999)
Phys. Rev. D 62, 081701 (2000).
P. Sutcliffe, Proc. Roy. Soc. Lond. A 463, 3001 (2007).
E. Radu and M.S. Volkov, Phys. Rep. 468, No. 4, 101 (2008).
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).
A.M. Gavrilik, J. Phys. A 27, 91 (1994)
Nucl. Phys. B (Proc. Suppl.) 102, 298 (2001).
R.J. Finkelstein, Int. J. Mod. Phys. A 22, 4467 (2007).
E. Witten, Comm. Math. Phys. 121, 351 (1989).
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).
L.C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989).
A.J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989).
I.I. Kachurik, Ukr. J. Math. 50, 1055 (1998).
V. Kac and P. Cheung, Quantum Calculus (Springer, New York, 2002).
How to Cite
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.