Knot homologies and generalized quiver partition functions

TL;DR

This paper explores a conjectured link between generalized quiver partition functions and knot homology polynomials, interpreting quiver nodes as holomorphic curves and analyzing recursion relations related to knot invariants.

Contribution

It introduces a novel interpretation of quiver nodes as holomorphic curves in the conifold and connects quiver partition functions to knot homologies and recursion relations.

Findings

Generalized quiver partition functions relate to HOMFLY-PT polynomials.

Quiver nodes correspond to specific holomorphic curves.

Partition functions satisfy recursion relations akin to toric branes.

Abstract

We conjecture a relation between generalized quiver partition functions and generating functions for symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincar\'e 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 $\mathbb{C}^3$.