The replacements of signed graphs and Kauffman brackets of link families

Abstract

Let G be a signed graph. Let G ˆ be the graph obtained from G by replacing each edge e of G by a chain or a sheaf. In this paper we first establish a relation between the Q-polynomial of G ˆ [L.H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989) 105–127] and the W-polynomial of G [L. Traldi, A dichromatic polynomial for weighted graphs and link diagrams, Proc. Amer. Math. Soc. 106 (1989) 279–286; B. Bollobás, O. Riordan, A Tutte polynomial for colored graphs, Combin. Probab. Comput. 8 (1999) 45–93]. Then we derive two special dual cases from the relation, one of which has been obtained in [X. Jin, F. Zhang, The Kauffman brackets for equivalence classes of links, Adv. in Appl. Math. 34 (2005) 47–64]. Based on the one-to-one correspondence between signed plane graphs and link diagrams, and the correspondence between the Q-polynomial of a signed plane graph and the Kauffman bracket of a link diagram, we can compute the Kauffman bracket of the link diagram corresponding to G ˆ by means of the W-polynomial of G. In this way we investigate the Kauffman brackets of rational links as a typical link family, and obtain an explicit formula using transfer matrix approach. Finally we provide another link family to point out that the relation we built actually can be used to deal with Kauffman brackets of all such link families.