We consider the generating function of Euler characteristics of stratified Hilbert schemes of a locally planar curve. We show that it is a rational function, which, at least in the case of unibranch singularities, depends only on the topology near the singular points. It is well known that the topology may be characterized by the knot (or, in general, link) obtained by intersecting the curve with the three-sphere bounding a small neighborhood of a point; we conjecture that in fact our generating function is the HOMFLY polynomial of this link. For singularities which are topologically equivalent to $x^m = y^n$, for $m,n$ relatively prime — i.e., corresponding to torus knots, we verify the conjecture by explicit calculation. A certain specialization of our generating function gives the so-called BPS contributions of an isolated Gorenstein curve in Pandharipande-Thomas theory. These have been shown to vanish in genera below the geometric genus and above the arithmetic genus. It follows from our conjecture that these are positive in this range. This talk presents joint work with Alexei Oblomkov.