Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
Abstract
Let JrSU(n)(K;q) denote the reduced SU(n) quantum invariant of a zero-framed knot K, colored by the rth symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed n≥2 we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients Hk(n)(K;q)∈ℤ[q±1] such that JrSU(n)(K;q)=∑k=0r (∏i=0k−1{r−i}{r+n+i}) Hk(n)(K;q), where {m}=qm−q−m. The finite dual interpolation formula of Beliakova--Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro--Lê convention, we restrict the completed center to one-row colors. Completed Harish--Chandra reflection then yields inversion symmetry in the variable z, and integral descent through X=z+z−1 converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients Gk(K;A,q)∈ℤ[A±1,q±1]. The associated Newton coefficients are Laurent polynomials in A over ℚ(q) whose values at every geometric node A=qn, n≥2, lie in ℤ[q±1]. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY--PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.
Related
- Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
- Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms
- Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
- Estimates of Lucas series and expansions in non-integer bases
- Completion Fibres of Infinite Positive Newton Conformal Metrics Near Corners
BibTeX
@misc{fang2026cyclotomic,
title = {{Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion}},
author = {Honghuai Fang and Tian Zhou},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29585},
eprint = {2608.29585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29585},
}
RIS
TY - UNPB
TI - Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
AU - Honghuai Fang
AU - Tian Zhou
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29585
UR - https://arxiv.org/abs/2608.29585
AB - Let J_r^SU(n)(K;q) denote the reduced SU(n) quantum invariant of a zero-framed knot K, colored by the rth symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed n≥2 we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients H_k^(n)(K;q)∈ℤ[q^±1] such that J_r^SU(n)(K;q)=∑_k=0^r (∏_i=0^k−1{r−i}{r+n+i}) H_k^(n)(K;q), where {m}=q^m−q^−m. The finite dual interpolation formula of Beliakova--Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro--Lê convention, we restrict the completed center to one-row colors. Completed Harish--Chandra reflection then yields inversion symmetry in the variable z, and integral descent through X=z+z^−1 converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients G_k(K;A,q)∈ℤ[A^±1,q^±1]. The associated Newton coefficients are Laurent polynomials in A over ℚ(q) whose values at every geometric node A=q^n, n≥2, lie in ℤ[q^±1]. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY--PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.
ER -