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Donaldson-Thomas invariants, torus knots, and lattice paths

Panfil, Miłosz and Stošić, Marko and Sułkowski, Piotr (2018) Donaldson-Thomas invariants, torus knots, and lattice paths. Physical Review D, 98 (2). Art. No. 026022. ISSN 2470-0010. doi:10.1103/PhysRevD.98.026022.

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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.

Item Type:Article
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Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© 2018 Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. Received 16 May 2018; published 16 July 2018. We thank Adam Doliwa, Eugene Gorsky, Sergei Gukov, Piotr Kucharski, and Markus Reineke for their interest in this work, useful comments, and enlightening discussions. Parts of this work were done while M. S. and P. S. were visiting the Max-Planck Institute for Mathematics (Bonn, Germany), American Institute for Mathematics (San Jose, USA), and Isaac Newton Institute for Mathematical Sciences (Cambridge, UK). This work is supported by the ERC Starting Grant No. 335739 “Quantum fields and knot homologies,” funded by the European Research Council under the European Union’s Seventh Framework Programme and the Foundation for Polish Science. The work of M. S. was also partially supported by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) through the FCT Investigador Grant No. IF/00998/2015 and also by the Ministry of Education, Science, and Technological Development of the Republic of Serbia, Project No. 174012. M. P. acknowledges the support from the National Science Centre through the FUGA Grant No. 2015/16/S/ST2/00448.
Funding AgencyGrant Number
European Research Council (ERC)335739
Foundation for Polish ScienceUNSPECIFIED
Fundação para a Ciência e a Tecnologia (FCT)IF/00998/2015
Ministry of Education, Science, and Technological Development (Serbia)174012
National Science Centre (Poland)2015/16/S/ST2/00448
Issue or Number:2
Record Number:CaltechAUTHORS:20180716-150139725
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87893
Deposited By: Tony Diaz
Deposited On:16 Jul 2018 23:03
Last Modified:16 Nov 2021 00:21

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