Abstract
AbstractWe introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal $$L_K$$
L
K
, a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. 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 $${\mathbb {C}}^3$$
C
3
.
Funder
Ministerstwo Edukacji i Nauki
Knut och Alice Wallenbergs Stiftelse
Vetenskapsrådet
National Center of Competence in Research Quantum Science and Technology
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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