Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A RECURRENCE RELATION FOR THE JONES POLYNOMIAL

  • Berceanu, Barbu (Simion Stoilow Institute of Mathematics, Abdus Salam School of Mathematical Sciences GC University) ;
  • Nizami, Abdul Rauf (Division of Science and Technology University of Education)
  • Received : 2013.06.19
  • Published : 2014.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

Using a simple recurrence relation, we give a new method to compute the Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for the Jones polynomials. The method is used to estimate the degree of the Jones polynomials for some families of braids and to obtain general qualitative results.

Keywords

References

  1. E. Artin, Theory of braids, Ann. of Math.(2) 48 (1947), 101-126. https://doi.org/10.2307/1969218
  2. R. Ashraf and B. Berceanu, Recurrence relation for HOMFLY polynomial and rational specializations, arXiv:1003.1034v1[mathGT], 2010.
  3. R. Ashraf and B. Berceanu, Simple braids, arXiv:1003.6014v1[mathGT], 2010.
  4. S. Bigelow, Does the Jones polynomial detect the unknot?, arXiv:0012086vI, 2000.
  5. J. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., No. 82. Princeton University Press, 1974.
  6. J. Birman, On the Jones polynomial of closed 3-braids, Invent. Math. 81 (1985), 287-294. https://doi.org/10.1007/BF01389053
  7. O. T. Dasbach and S. Hougardy, Does the Jones polynomial detect unknottedness?, Experiment. Math. 6 (1997), no. 1, 51-56. https://doi.org/10.1080/10586458.1997.10504350
  8. S. Eliahou, L. Kauffman, and M. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003), no. 1, 155-169. https://doi.org/10.1016/S0040-9383(02)00012-5
  9. F. A. Garside, The braid groups and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235-254. https://doi.org/10.1093/qmath/20.1.235
  10. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103-111. https://doi.org/10.1090/S0273-0979-1985-15304-2
  11. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388. https://doi.org/10.2307/1971403
  12. V. F. R. Jones, The Jones Polynomial, Discrete Math. 294 (2005), 275-277. https://doi.org/10.1016/j.disc.2004.10.024
  13. L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407. https://doi.org/10.1016/0040-9383(87)90009-7
  14. W. B. R. Lickorish, An Introduction to Knot Theory, Springer-Verlag New York, Inc., 1997.
  15. H. R. Morton and H. B. Short, Calculating the 2-variable polynomial for knots presented as closed braids, J. Algorithms 11 (1990), no. 1, 117-131. https://doi.org/10.1016/0196-6774(90)90033-B
  16. K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187-194. https://doi.org/10.1016/0040-9383(87)90058-9
  17. A. R. Nizami, Fibonacci modules and multiple Fibonacci sequences, ARS Combinatoria CXIII (2014), 151-159.
  18. A. Ocneanu, A polynomial invariant for knots: A combinatorial and algebraic approach, Preprint MSRI, Berkeley, 1984.
  19. A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, arXiv:0606255vI, 2006.
  20. M. Takahashi, Explicit formulas for jones polynomials of closed 3-braids, Comment. Math. Univ. St. Paul 38 (1989), no. 2, 129-167.
  21. M. B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297-309. https://doi.org/10.1016/0040-9383(87)90003-6
  22. P. Traczyk, 3-braids with proportional Jones polynomials, Kobe J. Math. 15 (1998), no. 2, 187-190.
  23. Y. Yokota, Twisting formulae of the Jones polynomials, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 473-482. https://doi.org/10.1017/S0305004100070559

Cited by

  1. Unoriented knot polynomials of torus links as Fibonacci-type polynomials pp.1793-7183, 2018, https://doi.org/10.1142/S1793557119500530