AM  Vol.5 No.17 , October 2014
A Recursive Approach to the Kauffman Bracket
ABSTRACT
We introduce a simple recursive relation and give an explicit formula of the Kauffman bracket of two-strand braid link . Then, we give general formulas of the bracket of the sequence of links of three-strand braids . Finally, we give an interesting result that the Kauffman bracket of the three-strand braid link is actually the product of the brackets of the two-strand braid links and . Moreover, a recursive relation for is also given.

Cite this paper
Nizami, A. , Munir, M. , Saleem, U. and Ramzan, A. (2014) A Recursive Approach to the Kauffman Bracket. Applied Mathematics, 5, 2746-2755. doi: 10.4236/am.2014.517262.
References
[1]   Kauffman, L.H. (1987) State Models and the Jones Polynomial. Topology, 26, 395-407.
http://dx.doi.org/10.1016/0040-9383(87)90009-7

[2]   Jaeger, F. (1990) A Combinatorial Model for the Homy Polynomial. European Journal of Combinatorics, 11, 549-555.

[3]   Reshetikhin, N.Y. (1988) Quantized Universal Enveloping Algebras, the Yang-Baxter Equation and Invariants of Links, I and II. LOMI Reprints E-4-87 and E-17-87, Steklov Institute, Leningrad, USSR.

[4]   Reidemeister, K. (1948) Knot Theory. Chelsea Publ and Co., New York.

[5]   Artin, E. (1925) Theorie der ZÖ pfe. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 4, 27-72.
http://dx.doi.org/10.1007/BF02950718

[6]   Artin, E. (1947) Theory of Braids. Annals of Mathematics, 48, 101-126.
http://dx.doi.org/10.2307/1969218

[7]   Birman, J.S. (1974) Braids, Links, and Mapping Class Groups. Princeton University Press, Princeton.

[8]   Manturov, V.O. (2004) Knot Theory. Chapman and Hall/CRC, Boca Raton.
http://dx.doi.org/10.1201/9780203402849

[9]   Murasugi, K. (1996) Knot Theory and Its Applications. BirkhäUser, Boston.

[10]   Alexander, J. (1923) Topological Invariants of Knots and Links. Transactions of the American Mathematical Society, 20, 275-306.

[11]   Adams, C.C. (1994) The Knot Book. W H Freeman and Company, New York.

 
 
Top