Algebraic & Geometric Topology

Minimal surface representations of virtual knots and links

H A Dye and Louis H Kauffman

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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.

Article information

Algebr. Geom. Topol., Volume 5, Number 2 (2005), 509-535.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

virtual knots minimal surface representation bracket polynomial Kishino knot


Dye, H A; Kauffman, Louis H. Minimal surface representations of virtual knots and links. Algebr. Geom. Topol. 5 (2005), no. 2, 509--535. doi:10.2140/agt.2005.5.509.

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