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arXiv · 2108.12645

Knot homologies and generalized quiver partition functions

Tobias Ekholm·Piotr Kucharski·Pietro Longhi
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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 $\mathbb{C}^3$.

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Tobias Ekholm, Piotr Kucharski, Pietro Longhi. 2021-08-28. Knot homologies and generalized quiver partition functions. https://arxiv.org/abs/2108.12645

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