Duke Math. J. 167 (3), 397-447, (15 February 2018) DOI: 10.1215/00127094-2017-0030
Stavros Garoufalidis, Aaron D. Lauda, Thang T. Q. Lê
KEYWORDS: knots, HOMFLYPT polynomial, colored HOMFLYPT polynomial, MOY graphs, webs, ladders, skew-Howe duality, quantum groups, q-holonomic, superpolynomial, Chern–Simons theory, 57N10, 57M25
We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a -holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an superpolynomial of knots in -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 -holonomic.