• Українська
  • English

< | >

Current issue   Ukr. J. Phys. 2014, Vol. 58, N 7, p.673-676
https://doi.org/10.15407/ujpe58.07.0673    Paper

Pavlyuk A.M.

Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: pavlyuk@bitp.kiev.ua)

Generalization of Polynomial Invariants and Holographic Principle for Knots and Links

Section: General problems of theoretical physics
Original Author's Text: English

Abstract: We formulate the holographic principle for knots and links. For the “space” of all knots and links, torus knots T(2m + 1, 2 ) and torus links L(2m, 2 ) play the role of the “boundary” of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for polynomial invariants of torus knots T(2m + 1, 2 ) and torus links L(2m, 2 ). As an example of the application of this principle, we derive the Jones skein relation and its generalization with the help of some variants of (q, p)-numbers, related with (q, p)-deformed bosonic oscillators.

Key words: holographic principle, knots, links, Jones skein relation.

References:

  1. The Interface of Knots and Physics, edited by L.H. Kauffman (Amer. Math. Soc., Providence, RI, 1995).
  2. L.H. Kauffman, Knots and Physics (World Scientific, Singapore, 2001).
     https://doi.org/10.1142/4256
  3. M.F. Atiyah, The Geometry and Physics of Knots (Cambridge Univ. Press, Cambridge, 1990).
  4. E. Witten, Comm. Math. Phys. 121, 351 (1989).
     https://doi.org/10.1007/BF01217730
  5. Knots and Quantum Gravity, edited by J. Baez (Oxford Univ. Press, Oxford, 1994).
  6. J. Baez and J.P. Muniain, Gauge Fields, Knots and Gravity (World Scientific, Singapore, 1994).
     https://doi.org/10.1142/2324
  7. G.'t Hooft, in Proceedings of the Salamfest (ICTP, Trieste, 1993), p. 283; gr-qc/9310026.
  8. J. Maldacena, Adv. Theor. Math. Phys. 2, 231-252 (1989); hep-th/9802150.
  9. E. Verlinde, JHEP 1104, 029 (2011); arXiv:1001.0785 [hep-th].
  10. A.M. Gavrilik and A.M. Pavlyuk, Ukr. J. Phys. 55, 129 (2010); arXiv:0912.4674v2 [math-ph].
  11. A.M. Pavlyuk, Algebras, Groups and Geometries 29, 151 (2012).
  12. A. Chakrabarti and R. Jagannathan, J. Phys. A: Math. Gen. 24, L711 (1991).
     https://doi.org/10.1088/0305-4470/24/13/002
  13. V.F.R. Jones, Bull. AMS 12, 103 (1985).
     https://doi.org/10.1090/S0273-0979-1985-15304-2
  14. P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millet, and A. Ocneanu, Bull. AMS 12, 239 (1985).
     https://doi.org/10.1090/S0273-0979-1985-15361-3
  15. A.M. Gavrilik and A.M. Pavlyuk, Ukr. J. Phys. 56, 680 (2011); arXiv:1107.5516v1 [math-ph].
  16. A.M. Pavlyuk, Ukr. J. Phys. 57, 439 (2012).