Minimal surface representations of virtual knots and links

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot diagram corresponds (up to generalized Reidemeister moves) to a unique embedding in a thickened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result and a generalised bracket polynomial, we develop methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial). As examples we show that, except for special cases, link diagrams with a single virtualization and link diagrams with a single virtual crossing are non-classical.

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Keywords

virtual knots, minimal surface representation, bracket polynomial, Kishino knot

Mathematical Subject Classification 2000

Primary: 57M25, 57M27

Secondary: 57N05

References

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Publication

Received: 31 May 2004

Accepted: 16 April 2005

Published: 4 June 2005

Authors

H A Dye
MADN-MATH United States Military Academy 646 Swift Road West Point NY 10996 USA

Louis H Kauffman
Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 South Morgan Street Chicago IL 60607-7045 USA

Volume 5, issue 2 (2005)

H A Dye and Louis H Kauffman

Abstract