What a knot sees at a root of unity
Abstract
HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters q and A. They also depend on the knot and representation of the gauge group. However, when q2 is a primitive m-th root of unity, e.g. q=± eiπ/m, these polynomials exhibit a universal (knot-independent) factorization: HR becomes the product of the polynomials for the m-core of the Young diagram R and of one and the same factor H[m] for every m-ribbon of R. In all the examples which we computed this factor is just the special polynomial at Am. At q=±1 factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At m>1 the relevant singularity is in the projector, and what survives of it is a finite sum of braids Xm -- the m-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.
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BibTeX
@misc{kononov2026what,
title = {{What a knot sees at a root of unity}},
author = {Ya. Kononov and A. Morozov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08242},
eprint = {2610.08242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08242},
}
RIS
TY - UNPB
TI - What a knot sees at a root of unity
AU - Ya. Kononov
AU - A. Morozov
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08242
UR - https://arxiv.org/abs/2610.08242
AB - HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters q and A. They also depend on the knot and representation of the gauge group. However, when q^2 is a primitive m-th root of unity, e.g. q=± e^iπ/m, these polynomials exhibit a universal (knot-independent) factorization: H_R becomes the product of the polynomials for the m-core of the Young diagram R and of one and the same factor H_[m] for every m-ribbon of R. In all the examples which we computed this factor is just the special polynomial at A^m. At q=±1 factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At m>1 the relevant singularity is in the projector, and what survives of it is a finite sum of braids X_m -- the m-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.
ER -