The Relationship Between The Kauffman Bracket Polynomials and The Tutte Polynomials of (2,n)-Torus Knots

Year 2017, Volume: 13 Issue: 4, 929 - 932, 29.12.2017

Abdulgani Şahin , Abdullah Kopuzlu

https://doi.org/10.18466/cbayarfbe.323841

Abstract

In knot theory there are many important invariants
that are hard to calculate. They are classified as numeric, group and
polynomial invariants. These invariants contribute to the problem of
classification of knots. In this study, we have done a study on the polynomial invariants
of the knots. First of all, for (2,n)-torus knots which is a special class of knots,
we calculated their the Kauffman bracket polynomials. We have found a general
formula for these calculations. Then the Tutte polynomials of signed graphs of
(2,n)-torus knots, marked with a {+} or {-} sign each on edge, have been
computed. Some results have been obtained at the end of these calculations. While
these researches have been studied, figures and regular diagrams of knots have
been applied so much. During the first calculation, we have used skein diagrams
and relations of the Kauffman polynomial. In the second calculation, the Tutte
polynomials of (2,n)-torus knots have been computed and at the end of the operation
some general formulas have been introduced. The signed graphs of (2,n)-torus
knots have been obtained by using their regular diagrams. Then the Tutte
polynomials of these graphs have been computed as a diagrammatic by recursive
formulas that can be defined by deletion-contraction operations. Finally, it
has been obtained that there is a relation between the Tutte polynomials and the
Kauffman bracket polynomials of (2,n)-torus knots.

Keywords

Kauffman polynomial, Knots, Knot graph, Torus knots

References

• [1] Murasugi, K. Knot Theory and Its Applications; Birkhau-ser Verlag, Boston, 1996; 337 pp.
• [2] Haggard G; Pearce D. J.; Royle G. Computing Tutte Polynomials. ACM Transactions on Mathematical Software 2010; 37(3), Article Number: 24.
• [3] Kauffman, L. H. A Tutte Polynomial for Signed Graphs. Discrete Applied Mathematics (1989); 25, 105–127.
• [4] Kopuzlu, A.; Şahin, A.; Uğur T. On Polynomials of (2,n)-Torus Knots. Applied Mathematical Sciences. HIKARI Ltd. 2009; 3(59), 2899 – 2910.
• [5] Şahin, A.; Kopuzlu A.; Uğur T. On Tutte Polynomials of (2,n)-Torus Knots. Applied Mathematical Sciences. HIKARI Ltd. 2015; 9(15), 747 – 759.
• [6] Burde, G.; Zieschang H. Knots; Walter De Gruyter, Ber-lin-New York 2003; 401 pp.
• [7] Cromwell, P. Knots and Links; Cambridge University Press New York, USA 2004; 328 pp.
• [8] Bollobas, B. Modern Graph Theory; Springer Science + Business Media, Inc, New York, USA 1998; 394 pp.
• [9] Tutte, W.T. A Contribution to The Theory of Chromatic Polynomials. Canadian Journal of Mathematics 1954; 6, 80-91.