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http://dx.doi.org/10.4134/JKMS.2014.51.3.443

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)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.3, 2014 , pp. 443-462 More about this Journal
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
Jones polynomial; braids; recurrence relation;
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1 V. F. R. Jones, The Jones Polynomial, Discrete Math. 294 (2005), 275-277.   DOI   ScienceOn
2 F. A. Garside, The braid groups and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235-254.   DOI
3 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.   DOI
4 V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388.   DOI   ScienceOn
5 K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187-194.   DOI   ScienceOn
6 L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407.   DOI   ScienceOn
7 W. B. R. Lickorish, An Introduction to Knot Theory, Springer-Verlag New York, Inc., 1997.
8 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.   DOI
9 A. R. Nizami, Fibonacci modules and multiple Fibonacci sequences, ARS Combinatoria CXIII (2014), 151-159.
10 A. Ocneanu, A polynomial invariant for knots: A combinatorial and algebraic approach, Preprint MSRI, Berkeley, 1984.
11 A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, arXiv:0606255vI, 2006.
12 M. Takahashi, Explicit formulas for jones polynomials of closed 3-braids, Comment. Math. Univ. St. Paul 38 (1989), no. 2, 129-167.
13 M. B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297-309.   DOI   ScienceOn
14 P. Traczyk, 3-braids with proportional Jones polynomials, Kobe J. Math. 15 (1998), no. 2, 187-190.
15 E. Artin, Theory of braids, Ann. of Math.(2) 48 (1947), 101-126.   DOI
16 J. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., No. 82. Princeton University Press, 1974.
17 R. Ashraf and B. Berceanu, Recurrence relation for HOMFLY polynomial and rational specializations, arXiv:1003.1034v1[mathGT], 2010.
18 R. Ashraf and B. Berceanu, Simple braids, arXiv:1003.6014v1[mathGT], 2010.
19 S. Bigelow, Does the Jones polynomial detect the unknot?, arXiv:0012086vI, 2000.
20 O. T. Dasbach and S. Hougardy, Does the Jones polynomial detect unknottedness?, Experiment. Math. 6 (1997), no. 1, 51-56.   DOI
21 S. Eliahou, L. Kauffman, and M. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003), no. 1, 155-169.   DOI   ScienceOn
22 J. Birman, On the Jones polynomial of closed 3-braids, Invent. Math. 81 (1985), 287-294.   DOI
23 Y. Yokota, Twisting formulae of the Jones polynomials, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 473-482.   DOI