Algebraic and Geometric Topology 4 (2004), paper no. 46, pages 1083-1101.

Span of the Jones polynomial of an alternating virtual link

Naoko Kamada

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.

Keywords. Virtual knot theory, knot theory

AMS subject classification. Primary: 57M25. Secondary: 57M27.

DOI: 10.2140/agt.2004.4.1083

E-print: arXiv:math.GT/0412074

Submitted: 4 March 2004. (Revised: 24 October 2004.) Accepted: 3 November 2004. Published: 21 November 2004.

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Naoko Kamada
Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku
Osaka, 558-8585, Japan

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