  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.
