  Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende,
relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial
of the associated algebraic link. More generally, we prove an extension of this
conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on
the conifold to the colored HOMFLY polynomial of the algebraic link. Our proof
uses wall-crossing techniques to prove a blowup identity on the
algebro-geometric side. We prove a matching identity for the colored HOMFLY
polynomials of a link using skein-theoretic techniques.
