Volume 5, issue 2 (2005)

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ISSN (electronic): 1472-2739
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Minimal surface representations of virtual knots and links

H A Dye and Louis H Kauffman

Algebraic & Geometric Topology 5 (2005) 509–535

arXiv: math.GT/0401035


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.

virtual knots, minimal surface representation, bracket polynomial, Kishino knot
Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 57N05
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Received: 31 May 2004
Accepted: 16 April 2005
Published: 4 June 2005
H A Dye
United States Military Academy
646 Swift Road
West Point NY 10996
Louis H Kauffman
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 South Morgan Street
Chicago IL 60607-7045