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2005 Minimal surface representations of virtual knots and links
H A Dye, Louis H Kauffman
Algebr. Geom. Topol. 5(2): 509-535 (2005). DOI: 10.2140/agt.2005.5.509

Abstract

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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H A Dye. Louis H Kauffman. "Minimal surface representations of virtual knots and links." Algebr. Geom. Topol. 5 (2) 509 - 535, 2005. https://doi.org/10.2140/agt.2005.5.509

Information

Received: 31 May 2004; Accepted: 16 April 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1083.57007
MathSciNet: MR2153118
Digital Object Identifier: 10.2140/agt.2005.5.509

Subjects:
Primary: 57M25, 57M27
Secondary: 57N05

Rights: Copyright © 2005 Mathematical Sciences Publishers

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