Minimal surface representations of virtual knots and links

Dye, H A¹ ; Kauffman, Louis H²

¹MADN-MATH, United States Military Academy, 646 Swift Road, West Point NY 10996, USA
²Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago IL 60607-7045, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 509-535

Résumé

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.

Détail

DOI : 10.2140/agt.2005.5.509

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

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