  We define three different types of spanning surfaces for knots in thickened
surfaces. We use these to introduce new Seifert matrices, Alexander-type
polynomials, genera, and a signature invariant. One of these Alexander
polynomials extends to virtual knots and can obstruct a virtual knot from being
classical. Furthermore, it can distinguish a knot in a thickened surface from
its mirror up to isotopy. We also propose several constructions of Heegaard
Floer homology for knots in thickened surfaces, and give examples why they are
not stabilization invariant. However, we can define Floer homology for virtual
knots by taking a minimal genus representative. Finally, we use the
Behrens-Golla $\delta$-invariant to obstruct a knot from being a stabilization
of another.
