Colored HOMFLYPT counts holomorphic curves

Abstract

Given a link in the three-sphere, its Lagrangian conormal can be transplanted to the “resolved conifold,” which is a certain noncompact Calabi–Yau threefold. Here we show that, as predicted by Ooguri and Vafa using string theoretic arguments, the count of all holomorphic curves in the resolved conifold ending on this Lagrangian is, appropriately understood, the collection of the HOMFLYPT invariants of all colorings of the link. This generalizes our previous work, Skeins on branes, arXiv:1901.08027, which identified a curve count that captures the uncolored case. The main ingredient in the present work is a skein-valued multiple cover formula for an isolated embedded annulus.