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(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?

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    $\begingroup$ Here is a simple comment: each two-component link can be realized in this way. In fact, each knot in $S^3$ bounds an immersed disk with some clasp singularities. You can choose one clasp disk for each component respectively, and perturb them into general position. By taking a band sum of them you will obtain the desired immersed disk. For links with components more than 2 this is also correct. $\endgroup$
    – Zhiyun Cheng
    Mar 20, 2016 at 4:46
  • $\begingroup$ That sounds very interesting, do you have a reference for this construction? $\endgroup$
    – Herng Yi
    Apr 4, 2016 at 17:38

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