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Thistlethwaite’s theorem for virtual links. (English)
J. Knot Theory Ramifications 17, No. 10, 1189-1198 (2008).
Shortly after the Jones polynomial $V(t)$ was introduced [{\it V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103‒111 (1985; Zbl 0564.57006)], {\it L. H. Kauffman} [Topology 26, 395‒407 (1987; Zbl 0622.57004)] gave his bracket description of $V(t)$ and {\it M. B. Thistlethwaite} [Topology 26, 297‒309 (1987; Zbl 0622.57003)] showed that the bracket may be obtained from the Tutte polynomial of a graph constructed using a checkerboard coloring of the complementary regions of a link diagram. In the paper under review the authors extend this machinery to Kauffman’s virtual links [{\it L. H. Kauffman}, Eur. J. Comb. 20, No. 7, 663‒690 (1999; Zbl 0938.57006)], by showing how to construct a ribbon graph from a virtual link diagram so that the Bollobás-Riordan topological Tutte polynomial of the ribbon graph [{\it B. Bollobás} and {\it O, Riordan}, Math. Ann. 323, No. 1, 81‒96 (2002; Zbl 1004.05021)] yields the Kauffman bracket of the diagram.
Reviewer: Lorenzo Traldi (Easton)