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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?

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If the knot moves by "regular isotopy" then the immersed surface it traces out is a torus. Thus the genus you desire is always one.

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