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Quantum Color Newton Rings

  • Honghuai Fang
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Abstract

We study integral Newton rings attached to symmetric quantum colors. We show that their boundary values at a deleted node are exactly the elements of Habiro's cyclotomic completion, and we describe their quantum neighborhoods at q=1 uniformly in the rank. The symmetric-color sln invariants of a knot are the values of an element of such a ring. In its expansion at q=1, the coefficient of (q−1)m has denominator dividing the (n−1+2m)th power of the Alexander polynomial, with integral numerator, and this exponent is sharp. For 52 and 61 at q=i, the period matrices of the rank-two coefficient recurrences have spectral modules of exact order three in the Picard group of an affine elliptic curve.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026quantum,
  title = {{Quantum Color Newton Rings}},
  author = {Honghuai Fang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21102},
  eprint = {2610.21102},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21102},
}
RIS
TY  - UNPB
TI  - Quantum Color Newton Rings
AU  - Honghuai Fang
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21102
UR  - https://arxiv.org/abs/2610.21102
AB  - We study integral Newton rings attached to symmetric quantum colors. We show that their boundary values at a deleted node are exactly the elements of Habiro's cyclotomic completion, and we describe their quantum neighborhoods at q=1 uniformly in the rank. The symmetric-color sl_n invariants of a knot are the values of an element of such a ring. In its expansion at q=1, the coefficient of (q−1)^m has denominator dividing the (n−1+2m)th power of the Alexander polynomial, with integral numerator, and this exponent is sharp. For 5_2 and 6_1 at q=i, the period matrices of the rank-two coefficient recurrences have spectral modules of exact order three in the Picard group of an affine elliptic curve.
ER  -