Volume 4, issue 2 (2004)

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Span of the Jones polynomial of an alternating virtual link

Naoko Kamada

Algebraic & Geometric Topology 4 (2004) 1083–1101

arXiv: math.GT/0412074


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.

virtual knot theory, knot theory
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
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Received: 4 March 2004
Revised: 24 October 2004
Accepted: 3 November 2004
Published: 21 November 2004
Naoko Kamada
Department of Mathematics
Osaka City University