Authors: Panfil, Miłosz
Stošić, Marko 
Sułkowski, Piotr
Title: Donaldson-Thomas invariants, torus knots, and lattice paths
Journal: Physical Review D
Volume: 98
Issue: 2
Issue Date: 15-Jul-2018
Rank: M21
ISSN: 2470-0010
DOI: 10.1103/PhysRevD.98.026022
In this paper, we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory - we find explicit formulas for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to (r,s) torus knots and show that their classical generating functions, in the extremal limit and framing rs, are generating functions of lattice paths under the line of the slope r/s. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and q-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schröder paths.
Publisher: American Physical Society
Project: Quantum fields and knot homologies 
Portuguese Fundação para a Ciência e a Tecnologia (FCT), FCT Investigador Grant No. IF/00998/2015
Geometry, Education and Visualization With Applications 
National Science Centre, FUGA Grant No. 2015/16/S/ST2/00448

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