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

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.