# Knot homologies and generalized quiver partition functions

Tobias Ekholm, Piotr Kucharski, Pietro Longhi

Tobias Ekholm, Piotr Kucharski, Pietro Longhi

**28/8/21**Published in : arXiv:2108.12645

We conjecture a relation between generalized quiver partition functions and generating functions for symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials of a knot K. We interpret the generalized quiver nodes as certain basic holomorphic curves with boundary on the knot conormal L_K in the resolved conifold, and the adjacency matrix as measuring their boundary linking. The simplest such curves are embedded disks with boundary in the primitive homology class of L_K, other basic holomorphic curves consists of two parts: an embedded punctured sphere and a multiply covered punctured disk with boundary in a multiple of the primitive homology class of L_K. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in \mathbb{C}^3.