Minimal surface representations of virtual knots and links

Kuperberg [15] 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.