(Migrated from Math Stack Exchange)

A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) A consequence of this is that two disjoint circles $C_1, C_2 \subset D$ cannot link in their image, that is, $f(C_1)$ is not linked with $f(C_2)$.

However if $f$ is an immersion then linking is enabled near a double curve: find two small disks inside $D$ whose images intersect along a double curve, then perturb them until the boundaries of those disks form a Hopf link.

The enablement of linking via an immersion seems interesting but is there any established theory on this that you could refer me to?