Minimal surface representations of virtual knots and links

Details Minimal surface representations of virtual knots and links Minimal surface representations of virtual knots and links

2004-01-05

arxiv.org/abs/math/0401035v5

Algebr. Geom. Topol. 5 (2005) 509-535

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-22.abs.html Version 5: a m

Scientific paper

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 thichened 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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