# Virtual knot diagrams on surfaces with genus?

To the best of my limited understanding, a virtual knot diagram may be thought of as the projection of an embedding of $\mathbb{S}^1$ in a 2-manifold with genus onto $\mathbb{R}^2$. That is to say it is an extension of the notion of a classical knot diagram to two dimensional handlebodies allowing for "virtual crossings" wherein one part of the diagram may "cross" another part by passing on the other side of a handle.

My question, then, is as follows:

What sort of object would be represented by a diagram with virtual crossings drawn on such a manifold? For example: if one were to draw a virtual knot diagram on $\mathbb{T}^2$ in a non-trivial manner (i.e. such that the diagram is drawn in a neighborhood of the torus that is not homeomorphic to an open disk in $\mathbb{R}^2$), what would this diagram be the projection of?

Note: If I have misused any terminology please feel free to correct me. (I have yet to take any courses on topology)

-