The colored HOMFLYPT function is $q$-holonomic

TL;DR

This paper proves that the colored HOMFLYPT polynomial is a $q$-holonomic function, establishing the existence of a super-polynomial for knots, using skew Howe duality and quantum group computations.

Contribution

It demonstrates the $q$-holonomicity of the HOMFLYPT polynomial for links colored by partitions with fixed rows, confirming a conjecture related to knot super-polynomials.

Findings

HOMFLYPT polynomial is $q$-holonomic for fixed-row colorings

Existence of an $(a,q)$ super-polynomial for knots confirmed

Proof uses skew Howe duality and quantum group calculations

Abstract

We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed number of rows is a $q$-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ super-polynomial 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 Poincare-Birkhoff-Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram.