Span of the Jones polynomial of an alternating virtual link

Abstract

For an oriented virtual link, L H Kauffman defined the $f$–polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the $f$–polynomial of an alternating virtual link $L$ is determined by the number of crossings of any alternating diagram of $L$ and the supporting genus of the diagram. It is a generalization of Kauffman–Murasugi–Thistlethwaite’s theorem. We also prove a similar result for a virtual link diagram that is obtained from an alternating virtual link diagram by virtualizing one real crossing. As a consequence, such a diagram is not equivalent to a classical link diagram.