In this study, we introduce the relationship between the Tutte polynomials and dichromatic polynomials of (2,n)-torus knots. For this aim, firstly we obtain the signed graph of a (2,n)-torus knot, marked with {+} signs, via the regular diagram of its. Whereupon, we compute the Tutte polynomial for this graph and find a generalization through these calculations. Finally we obtain dichromatic polynomial lying under the unmarked states of the signed graph of the (2,n)-torus knots by the generalization.

#### Keywords

- Knot
- knot graph
- Tutte polynomial
- dichromatic polynomial
- signed graph

#### MSC 2010

- 57M15
- 57M25
- 57M27

William Tutte devised a renewed polynomial for graphs at the year of 1954 [9]. This polynomial has great importance in mathematics, statiscal physics, biology and theoretical computer science. It is a polynomial of two variable.

In 1984, for knots and links, a new polynomial invariant was described by Jones. For the Jones polynomial, Thistlethwaite acquired a spanning tree dilation in 1987 [8]. His study indicated that a Tutte polynomial of a connected plane graph equals the Jones polynomial of an alternating oriented link. Kauffman developed this result in 1988 and he defined Kauffman bracket polynomials [4].

Jaeger indicated that the Tutte polynomial of a plane graph equals homfly polynomial of a connected link in 1988 [2]. A characterization of the Tutte polynomial for signed graphs was carried out by Kauffman in 1989. It was shown with

A knot is defined as a simple closed curve. More conceptually, a knot is embedding of the circle ^{1} into ^{3} (or ^{3}) [6]. A torus knot can be constructed on the trivial torus without any intersection points. If this is possible it takes the name of the torus knot. As specified by Murasugi [6], “the torus knot of the sort of (_{(p,q)} such that

A graph

The set of associated plane link diagrams and the set of associated signed plane graphs correspond to each other individually [5].

_{p}_{p}_{n}_{n}_{p}_{p}_{n}_{n}

In the graph ^{′} and ^{″} are graphs obtained via deletion-contraction.

If

Providing _{1} and _{2} at the time,

Tutte also described the dichromatic polynomial of a graph as a closer two-variable generalization of the chromatic polynomial. It is

The Tutte polynomial of a graph can be reformulated to the dichromatic polynomial of a graph. For this reason,

(2,n)-torus knots indicated by _{2,n} are considered. Firstly, it is obtained the isomorphic graphs belonging to (2,n)-torus knots _{2,n} by constructing regular diagrams of them. After that, (+) or (−) signs are assigned to these isomorphic graphs belonging to (2,n)-torus knots according to a certain rule. And then their Tutte polynomials reckon explained as above. [7] can be viewed for more details. We will debate over these results.

The graph signed with (+) of

By using the general formula in Theorem 2, the relation in proposition 1 and the abbreviation equations

Here we get a characterization for graphs of (2,n)-torus knots. In doing so, we are establishing an organic bond between the knot and its graph. We propose an alternative way of calculating the dichromatic polynomial. We are concretizing what is done with an example. In the following stages, the relation of this study with other graph polynomials is a worthy problem.

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