Shading A-polynomials via huge representations of Uq(suN)
Abstract
Classical A-polynomials A(ℓ,m) define constraints on coordinates ℓ and m in SL(2,ℂ) (a complexification of SU(2)) character varieties associated to knot complements S3∖ K. Quantum A-polynomials Â(ℓ̂,m̂) are difference operators annihilating Jones polynomials believed to represent wave functions of 3d Chern-Simons theory with gauge group SU(2) on a toroidal pipe surrounding the knot K strand -- a boundary of the knot complement. We suggest a construction of classical shaded A-polynomials Aa(ℓb,mc) associated to Lie groups SU(N), assuming that color N adds some extra "depth" and "shade" to the classical notion of A-polynomials. We exploit a formalism of Clebsch-Gordan (CG) chords, where indices a, b, c run over 1,…,N−1. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or in terms of quantum group Uq(suN). In the case of su2, CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analogue of A-polynomials, known as augmentation polynomials. The CG chord formalism can be easily extended to arbitrary suN allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary suN and arbitrary representations as well. We primarily aim at classical A-polynomials by considering a double scaling limit in which q=eℏ, ℏ→ 0 and the representations are huge: highest weight vector components wi→ ∞ while combinations ℏ wi∼ mi remain finite. We expect the presented techniques will be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras g. We discuss explicit examples of A-polynomials for knots 31, 41 and 51 for g=su3.
Versions
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BibTeX
@misc{galakhov2026shading,
title = {{Shading A-polynomials via huge representations of $U_q(\mathfraksu_N)$}},
author = {Dmitry Galakhov and Alexei Morozov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.22560},
eprint = {2605.22560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.22560},
}
RIS
TY - UNPB
TI - Shading A-polynomials via huge representations of U_q(su_N)
AU - Dmitry Galakhov
AU - Alexei Morozov
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2605.22560
UR - https://arxiv.org/abs/2605.22560
AB - Classical A-polynomials A(ℓ,m) define constraints on coordinates ℓ and m in SL(2,ℂ) (a complexification of SU(2)) character varieties associated to knot complements S^3∖ K. Quantum A-polynomials Â(ℓ̂,m̂) are difference operators annihilating Jones polynomials believed to represent wave functions of 3d Chern-Simons theory with gauge group SU(2) on a toroidal pipe surrounding the knot K strand -- a boundary of the knot complement. We suggest a construction of classical shaded A-polynomials A_a(ℓ_b,m_c) associated to Lie groups SU(N), assuming that color N adds some extra "depth" and "shade" to the classical notion of A-polynomials. We exploit a formalism of Clebsch-Gordan (CG) chords, where indices a, b, c run over 1,…,N−1. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or in terms of quantum group U_q(su_N). In the case of su_2, CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analogue of A-polynomials, known as augmentation polynomials. The CG chord formalism can be easily extended to arbitrary su_N allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary su_N and arbitrary representations as well. We primarily aim at classical A-polynomials by considering a double scaling limit in which q=e^ℏ, ℏ→ 0 and the representations are huge: highest weight vector components w_i→ ∞ while combinations ℏ w_i∼ m_i remain finite. We expect the presented techniques will be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras g. We discuss explicit examples of A-polynomials for knots 3_1, 4_1 and 5_1 for g=su_3.
ER -