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

Yıl 2017, Cilt: 13 Sayı: 4, 929 - 932, 29.12.2017

Abdulgani Şahin , Abdullah Kopuzlu

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

Öz

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.

Anahtar Kelimeler

Kauffman polynomial, Knots, Knot graph, Torus knots

Kaynakça

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