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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)

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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.

The connect sum of virtual knots is not well defined, for example, look up Kishino's knot. The source for this answer was Manturov's book called "Virtual Knots; the state of the art", which is a rather heavy read I cannot recommend if you still haven't taken topology.

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