# 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)

this is a very interesting question. Your understanding is pretty accurate, though you do not technically need to represent the diagram in $\mathbb R^2$ to obtain a virtual knot. The embedding $K:\mathbb S^1 \to M\times I$ , where $M$ is a 2-manifold with genus g, and $I$ is the unit interval (the cartesian product yields a thickened surface) is enough to make $K$ a virtual knot.
You might notice I do not require $g>1$, since the beauty of virtual knot theory is that it is a natural extension of classical knot theory. Anyways, your idea can corresponds to a preexisting operation called connect sum, where you replace a part of the virtual knot that doesn't have crossings with a $(1,1)$ virtual tangle. This approach is entirely diagram based. In geometrical terms, you have $K_1 \subset G_1\times I$ and $K_2 \subset G_2\times I$, two virtual knots, and you identify open balls in each thickened surface such that $\partial\overline B\cap K_i$ are two points for each knot and they are pairwise identified.