The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G_D. However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F_G and S_G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.