A nice softie "application" (?connection?) that I like is the
observation that the space of subsets of $S^1$ containing at most
three elements is homeomorphic to $S^3$. The space of subsets of
$S^1$ with at most $3$ elements is usually denoted $\exp_3 S^1$.
$$\exp_3 S^1 \simeq S^3$$
$SO_2$ acts on $\exp_3 S^1$ by rotation. The action is fixed-point
free. Consider its orbit stratification. There are the free orbits,
the orbits with isotropy $\mathbb Z_2$ and orbits with isotropy
$\mathbb Z_3$, and that's it.
This is enough information to deduce that $\exp_3 S^1$ as an $SO_2$
space is equivalent to $S^3 \subset \mathbb C^2$ as an $SO_2$-space
with the action:
$$ SO_2 \times \mathbb C^2 \to \mathbb C^2$$
$$ (\alpha, (z_1,z_2)) \longmapsto (\alpha^3z_1,\alpha^2z_2)$$
here I'm thinking of $SO_2$ as the unit complex numbers.
Anyhow, the "high point" of this cycle of observations is that the
free orbits of this action on $S^3$ are trefoil
Depending on what path you take maybe you're not using any knot theory
at all for this, but the fact that a non-trivial knot arises naturally
IMO is pretty cool. I believe the trefoil comes up in a few other
branches of mathematics quite naturally.
(edit: earlier I said symmetric product. Technically the symmetric
product is $(S^1)^3/\Sigma_3 \simeq S^3$ is homeomorphic to $D^2
\times S^1$. There is a natural onto map $(S^1)^3 / \Sigma_3 \to
\exp_3 S^1$ which collapses the boundary torus to a Moebius band)