Algebraic and Geometric Topology 5 (2005), paper no. 22, pages 509-535.

Minimal surface representations of virtual knots and links

H.A. Dye, Louis H. Kauffman

Abstract. Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot 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.

Keywords. Virtual knots, minimal surface representation, bracket polynomial, Kishino knot

AMS subject classification. Primary: 57M25, 57M27. Secondary: 57N05.

DOI: 10.2140/agt.2005.5.509

E-print: arXiv:math.GT/0401035

Submitted: 31 May 2004. Accepted: 16 April 2005. Published: 4 June 2005.

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H.A. Dye, Louis H. Kauffman
MADN-MATH, United States Military Academy
646 Swift Road, West Point, NY 10996, USA
and
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago, 851 South Morgan St
Chicago, IL 60607-7045, USA
Email: hdye@ttocs.org, kauffman@uic.edu

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