Applications of knot theory

If I may steal some thunder from Peter Shor, his paper, Quantum money from knots (with Edward Farhi, David Gosset, Avinatan Hassidim, and Andrew Lutomirski) relies for the security of its "quantum money scheme" on

the assumption that given two different looking but equivalent knots, it is difficult to explicitly find a transformation that takes one to the other.

The Alexander polynomial plays a prominent role in the paper.

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" and pursued further in Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities".

If you manage to convince your students that smooth manifolds are among the most beautiful and interesting objects in mathematics, expecially dimensions 3 and 4 that model our universe, then you can say that (among other things) knots form a fundamental ingredient in understanding and constructing such models.

For instance, you can tell that by removing a (well-chosen) knot from $S^3$ we can get the simplest possible universe with a hyperbolic geometry having finite volume. Or that every 3-manifold may be constructed by removing and "regluing" (finitely many) knots.