A 2-variable polynomial invariant for a virtual link derived from magnetic graphs
A virtual link is a link diagram in the plane possibly with some encircled crossings without over/under information, called virtual crossings. \textit{L. H. Kauffman} [Eur. J. Comb. 20, No. 7, 663--690 (1999; Zbl 0938.57006)] defined a polynomial invariant (called Jones-Kauffman polynomial) \(f_L(A)\in\mathbb{Z} [A^2,A^{-2}]\) for a virtual link \(L\) (for a classical link \(L\), it is the Jones polynomial after a suitable substitution). \(f_L(A)\) is decomposed nontrivially into two parts belonging to \(\mathbb{Z}[A^4,A^{-4}]\) and to \(\mathbb{Z}[A^4,A^{-4}]\cdot A^2\). In this paper, the authors propose a new 2-variable polynomial invariant for virtual links by using magnetic graphs (that are 2-regular graphs \(G\) in the 3-sphere such that the edges of \(G\) are oriented alternately) and show its relation to the latter splitting of the Jones-Kauffman polynomial.
scientific article; zbMATH DE number 5011030