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arXiv · 2610.21102

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 $\mathfrak{sl}_n$ 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.

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Honghuai Fang. 2026-10-15. Quantum Color Newton Rings. https://arxiv.org/abs/2610.21102

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