On the Jones Polynomial in the Solid Torus

Authors

  • Khaled Bataineh Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110

DOI:

https://doi.org/10.26713/cma.v7i4.510

Keywords:

Knots, Links, Jones polynomial, Solid torus

Abstract

We introduce an infinite class of elementary knots in the solid torus, together with general recursive and explicit formulas of the values of these knots under the generalized invariant of Jones polynomial to the solid torus. These values can be used as an infinite set of initial data for this invariant. We also introduce a procedure of resolving certain knots called spiral knots into these elementary knots. We show that our explicit formulas involve exactly \(n+1\) terms for an elementary knot with n crossings, which reduces the calculations needed to compute the invariant for spiral knots and arbitrary knots and links in the solid torus.

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References

K. Bataineh and M. Hajij, Jones polynomial for links in the handlebody, Rocky Mountain Journal of Mathematics, 43 (3) (2013), 737–753.

K. Bataineh and H. Belkhirat, The derivatives of the Hoste and Przytycki polynomial for oriented links in the solid torus, accepted for publication in Houston Journal of Mathematics.

P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bulletin of the American Mathematical Society, 12 (2) (1985), 239–246.

J. Hoste and M. Kidwell, Dichromatic link invariants, Trans. Amer. Math. Soc., 321 (1) (1990), 197–229.

J. Hoste and J. Przytycki, An invariant of dichromatic links, Proc. Amer. Math. Soc., 105 (1989), 1003–1007.

V. Jones, A polynomial invariant for knots via von Neumann algebra, Bull. Amer. Math. Soc., 12 (1985), 103–111.

L. Kauffman, New invariants in the theory of knots, American Mathematical Monthly, 95 (1988), 195–242.

A. Kawauchi, A Survey of Knot Theory, Birkhauser-Verlag (1996).

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Published

31-12-2016
CITATION

How to Cite

Bataineh, K. (2016). On the Jones Polynomial in the Solid Torus. Communications in Mathematics and Applications, 7(4), 291–302. https://doi.org/10.26713/cma.v7i4.510

Issue

Vol. 7 No. 4 (2016)

Section

Research Article

License

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