Duke Mathematical Journal

The colored HOMFLYPT function is q-holonomic

Stavros Garoufalidis, Aaron D. Lauda, and Thang T. Q. Lê

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We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a q-holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an (a,q) superpolynomial of knots in 3-space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincaré–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly q-holonomic.

Article information

Duke Math. J., Volume 167, Number 3 (2018), 397-447.

Received: 28 April 2016
Revised: 18 April 2017
First available in Project Euclid: 10 November 2017

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Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

knots HOMFLYPT polynomial colored HOMFLYPT polynomial MOY graphs webs ladders skew-Howe duality quantum groups q-holonomic superpolynomial Chern–Simons theory


Garoufalidis, Stavros; Lauda, Aaron D.; Lê, Thang T. Q. The colored HOMFLYPT function is $q$ -holonomic. Duke Math. J. 167 (2018), no. 3, 397--447. doi:10.1215/00127094-2017-0030. https://projecteuclid.org/euclid.dmj/1510304421

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