Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms
Abstract
Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen--Liu--Zhu for colored SU(n) invariants. We prove this conjecture for torus knots T(2,2p+1) and give a Bailey theoretic realization of the cyclotomic coefficients. For n≥2 and Kp=T(2,2p+1), the colored SU(n) invariants admit an expansion JNSU(n)(Kp;q) = ∑m=0N (∏j=0m−1{N−j}{N+n+j}) Hm(n,p)(q), where Hm(n,p)(q)∈ℤ[q±1] is independent of the color N. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin--Zheng formula for T(2,2p+1) then gives a well-poised Bailey kernel. A terminating very-well-poised 6ϕ5 summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral q-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen--Liu--Zhu SU(n) volume conjecture for T(2,2p+1).
Related
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BibTeX
@misc{wang2026cyclotomic,
title = {{Cyclotomic expansions of colored $SU(n)$ invariants of two-strand torus knots and Bailey transforms}},
author = {Chuwen Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13766},
eprint = {2609.13766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13766},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms
AU - Chuwen Wang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13766
UR - https://arxiv.org/abs/2609.13766
AB - Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen--Liu--Zhu for colored SU(n) invariants. We prove this conjecture for torus knots T(2,2p+1) and give a Bailey theoretic realization of the cyclotomic coefficients. For n≥2 and K_p=T(2,2p+1), the colored SU(n) invariants admit an expansion J_N^SU(n)(K_p;q) = ∑_m=0^N (∏_j=0^m−1{N−j}{N+n+j}) H_m^(n,p)(q), where H_m^(n,p)(q)∈ℤ[q^±1] is independent of the color N. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin--Zheng formula for T(2,2p+1) then gives a well-poised Bailey kernel. A terminating very-well-poised _6ϕ_5 summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral q-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen--Liu--Zhu SU(n) volume conjecture for T(2,2p+1).
ER -