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I think historically one of the big formal motivations for knot theory were things like the Brieskorn varieties, i.e. looking at solutions to equations of the form

$$z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_n} = 0$$

in $\mathbb C^n$, for various $p_k$. $0$ is generally a singular point and one way to study it is to intersect the variety with a small sphere centred about $0$. In the $n=2$ case you get knots in spheres, in many cases you get homology spheres (sometimes homotopy-spheres) knotted in spheres. The knot type informs on the singularity.

In the $n=3$ case you get the 3-dimensional Brieskorn varieties. By forgetting various coordinates this expresses the Brieskorn manifolds as branched covering spaces of $S^3$ branched over a knot. So again knots arrise. Looking at M.Epple's "Geometric aspects in the development of knot theory" apparently this perspective on branched coverings and knots goes back to Wirthinger (1895).

This perspective is well written-up in Milnor's "Singular points of complex hypersurfaces"hypersurfaces" and pursued further in Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities".

$$z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_n} = 0$$
in $\mathbb C^n$, for various $p_k$. $0$ is generally a singular point and one way to study it is to intersect the variety with a small sphere centred about $0$. In the $n=2$ case you get knots in spheres, in many cases you get homology (sometimes homotopy-spheres) knotted in spheres. The knot type informs on the singularity.
In the $n=3$ case you get the 3-dimensional Brieskorn varieties. By forgetting various coordinates this expresses the Brieskorn manifolds as branched covering spaces of $S^3$ branched over a knot. So again knots arrise. Looking at M.Epple's "Geometric aspects in the development of knot theory" apparently this perspective on branched coverings and knots goes back to Wirthinger (1895).