Virtual knot

A classical knot can be considered an ambient isotopy class of embeddings of the circle into a thickened 2-sphere.

This can be generalized by considering such classes of embeddings into thickened higher-genus surfaces.

This is not quite what we want since adding a handle to a (thick) surface will create a higher-genus embedding of the original knot.

The adding of a handle is called stabilization and the reverse process destabilization.

Thus a virtual knot can be considered an ambient isotopy class of embeddings of the circle into thickened surfaces with the equivalence given by (de)stabilization.