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 knots. http://www.msp.warwick.ac.uk/agt/2002/02/p043.xhtml

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)