Knots and graphs
This paper is an exposition of relatively recent results concerning the interplay between graph and knot theory. The authors start with two specific graphs (the knot graph used by Bankwitz, the Seifert graph) extracted from a knot or link diagram and present two invariants of a graph (the polynomial of a signed graph, the cycle index) and some of their properties, and then look at how these two graph invariants can be used to calculate the braid index of an oriented link (especially a \(2\)-bridge knot and an alternating knot) and the minimum crossing number of an alternating link and a torus knot, together with several examples. In particular, this paper contains two conjectures which can be found in another expository article [\textit{K.Murasugi}, Sugaku Expo. 9, No. 2, 187-205 (1996); translation from Sugaku 46, No. 2, 91-111 (1994; Zbl 0867.57008)]. The paper will be of value to researchers in this field, as a concise introduction to many results and their relevant works.
scientific article; zbMATH DE number 1370405
scientific article