# Genus of the surface traced out by a knot

I am no good at visualizing things, and to add to the misery, have only a passing acquaintance with knot theory, so at the risk of sounding silly I dare ask in loose terms: if I take a knot $$K$$ in $$\mathbb{R}^3$$ and "move it around in a circle until it comes back to itself without ever intersecting itself en route", thus inscribing a closed surface $$S(K)$$ (which would most likely be horribly self-intersecting), can the genus of the (singular) surface be expressed in terms of some invariant of the knot?

An example of the procedure hinted at above would be moving a circle around in a circle to get a (hollow) torus.

Formally I suppose $$S(K)$$ would be described as a $$K$$-fiber bundle over $$S^1$$, embedded in $$\mathbb{R}^3$$. So, rather than ask for its genus, one could better frame the question in terms of a suitable invariant of the bundle perhaps?