Knot homologies and generalized quiver partition functions

Ekholm, Tobias,1970-Uppsala universitet,Geometri och fysik,Inst Mittag Leffler, Aurav 17, S-18260 Djursholm, Sweden. (author)

We 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 Poincare polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal 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 C-3.

Kucharski, PiotrCALTECH, Walter Burke Inst Theoret Phys, Pasadena, CA 91125 USA.;Univ Warsaw, Fac Phys, Ul Pasteura 5, PL- 02093 Warsaw, Poland.;Univ Warsaw, Inst Math, Ul Banacha 2, PL-02097 Warsaw, Poland. (author)
Longhi, PietroSwiss Fed Inst Technol, Inst Theoret Phys, CH-8093 Zurich, Switzerland. (author)
Uppsala universitetGeometri och fysik (creator_code:org_t)

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NATURAL SCIENCES: NATURAL SCIENCES; and Physical Science ...; and Subatomic Physic ...

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