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15 February 2018 The colored HOMFLYPT function is q-holonomic
Stavros Garoufalidis, Aaron D. Lauda, Thang T. Q. Lê
Duke Math. J. 167(3): 397-447 (15 February 2018). DOI: 10.1215/00127094-2017-0030

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.

Citation

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Stavros Garoufalidis. Aaron D. Lauda. Thang T. Q. Lê. "The colored HOMFLYPT function is q-holonomic." Duke Math. J. 167 (3) 397 - 447, 15 February 2018. https://doi.org/10.1215/00127094-2017-0030

Information

Received: 28 April 2016; Revised: 18 April 2017; Published: 15 February 2018
First available in Project Euclid: 10 November 2017

zbMATH: 06848176
MathSciNet: MR3761103
Digital Object Identifier: 10.1215/00127094-2017-0030

Subjects:
Primary: 57N10
Secondary: 57M25

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

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 3 • 15 February 2018
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