## Duke Mathematical Journal

### The colored HOMFLYPT function is $q$-holonomic

#### Abstract

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

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

Dates
Revised: 18 April 2017
First available in Project Euclid: 10 November 2017

https://projecteuclid.org/euclid.dmj/1510304421

Digital Object Identifier
doi:10.1215/00127094-2017-0030

Mathematical Reviews number (MathSciNet)
MR3761103

Zentralblatt MATH identifier
06848176

#### Citation

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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