## Duke Mathematical Journal

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

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

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

#### 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**

Received: 28 April 2016

Revised: 18 April 2017

First available in Project Euclid: 10 November 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-2017-0030

**Mathematical Reviews number (MathSciNet)**

MR3761103

**Zentralblatt MATH identifier**

06848176

**Subjects**

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

**Keywords**

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

#### 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