Graph colourings and link invariants.

*(English)*Zbl 0693.05032
Graph colourings, Proc. Conf., Milton Keynes/UK 1988, Pitman Res. Notes Math. Ser. 218, 97-114 (1990).

[For the entire collection see Zbl 0693.05025.]

Some recent advances in knot theory involve combinatorial methods related to the chromatic theory of plane graphs. These advances consist of the discovery of new polynomial invariants of links and of their application to the solution of long-standing problems. The new invariants can be defined either algebraically or combinatorially using a plane projection of the link, called a diagram, which can be viewed as a graph. A nice connection between the algebraic and combinatorial aspects has been exhibited through the construction of models inspired by theoretical physics. Some of these models, related to the partition functions of statistical mechanics, can be viewed as weighted enumerators of certain colourings (in the broad sense) of the link diagram. This leads to interesting relationships with classical graph invariants such as the Tutte polynomial and the chromatic polynomial. In this paper one of the new invariants, the Kauffman polynomial is studied, and a special class of IRF (interaction round a face) models which provide a unifying frame for a number of combinatorial results concerning specializations of the Kauffman polynomial is presented. The author presents the Kauffman polynomial in Section 3, using the two versions which are most suitable for the computations. In Section 4 a class of IRF models are given, which are designed to coincide with the Kauffman polynomial, provided that a certain system of equations is satisfied. In Section 5 the solutions of this system are classified and they are related to some interesting special cases of the Kauffman polynomial: the generating function for the writhe of the orientations, the bracket polynomial, and its square. In the same process a result of M. B. Thistlethwaite [Invent. Math. 93, No.2, 285-296 (1988; Zbl 0645.57007)], relating the Kauffman and chromatic polynomials is obtained. Finally, in Section 6, some consequences and open questions are proposed.

Some recent advances in knot theory involve combinatorial methods related to the chromatic theory of plane graphs. These advances consist of the discovery of new polynomial invariants of links and of their application to the solution of long-standing problems. The new invariants can be defined either algebraically or combinatorially using a plane projection of the link, called a diagram, which can be viewed as a graph. A nice connection between the algebraic and combinatorial aspects has been exhibited through the construction of models inspired by theoretical physics. Some of these models, related to the partition functions of statistical mechanics, can be viewed as weighted enumerators of certain colourings (in the broad sense) of the link diagram. This leads to interesting relationships with classical graph invariants such as the Tutte polynomial and the chromatic polynomial. In this paper one of the new invariants, the Kauffman polynomial is studied, and a special class of IRF (interaction round a face) models which provide a unifying frame for a number of combinatorial results concerning specializations of the Kauffman polynomial is presented. The author presents the Kauffman polynomial in Section 3, using the two versions which are most suitable for the computations. In Section 4 a class of IRF models are given, which are designed to coincide with the Kauffman polynomial, provided that a certain system of equations is satisfied. In Section 5 the solutions of this system are classified and they are related to some interesting special cases of the Kauffman polynomial: the generating function for the writhe of the orientations, the bracket polynomial, and its square. In the same process a result of M. B. Thistlethwaite [Invent. Math. 93, No.2, 285-296 (1988; Zbl 0645.57007)], relating the Kauffman and chromatic polynomials is obtained. Finally, in Section 6, some consequences and open questions are proposed.

Reviewer: I.Tomescu

##### MSC:

05C15 | Coloring of graphs and hypergraphs |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |