A minimal surface is one whose area is stationary under small perturbations. If you dip a loop of wire in soapy water and pull it out, the soap film will naturally form a minimal surface. The Plateau problem is to find a minimal surface which bounds a given closed loop. From the point of view of analysis, this involves solving a non-linear PDE. I will explain how counting the number of solutions to a version of the Plateau problem can lead to knot invariants. Given a knot K in the three-sphere, one can count minimal surfaces in 4-dimensional hyperbolic space which are asymptotic at infinity to K and the answer does not change when you deform the knot. I will explain a conjecture that this count of minimal surfaces actually recovers the "HOMFLYPT" polynomial of the knot. This polynomial is purely combinatorial and simple to calculate from a planar diagram of the knot. If the conjecture is true, it would mean that easy combinatorial calculations can give existence of minimal surfaces! I will assume no knowledge of either minimal surfaces or knots. There will be lots of pictures