Span of the Jones polynomial of an alternating virtual link

Kamada, Naoko

doi:10.2140/agt.2004.4.1083

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Span of the Jones polynomial of an alternating virtual link

Naoko Kamada

Algebraic & Geometric Topology 4 (2004) 1083–1101

DOI: 10.2140/agt.2004.4.1083

arXiv: math.GT/0412074

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

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27

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Publication

Received: 4 March 2004

Revised: 24 October 2004

Accepted: 3 November 2004

Published: 21 November 2004

Authors

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

Volume 4, issue 2 (2004)