For an oriented virtual link, L H Kauffman defined the –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 –polynomial of an alternating virtual link is determined by the number of crossings of any alternating diagram of 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.
"Span of the Jones polynomial of an alternating virtual link." Algebr. Geom. Topol. 4 (2) 1083 - 1101, 2004. https://doi.org/10.2140/agt.2004.4.1083