Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion

Abstract

Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients $H_k^{(n)}(K;q)\in\mathbb{Z}[q^{\pm1}]$ such that \[ J_r^{SU(n)}(K;q)=\sum_{k=0}^{r} \left(\prod_{i=0}^{k-1}\{r-i\}\{r+n+i\}\right) H_k^{(n)}(K;q), \] 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--L\^e 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 $G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$. The associated Newton coefficients are Laurent polynomials in $A$ over $\mathbb{Q}(q)$ whose values at every geometric node $A=q^n$, $n\ge2$, lie in $\mathbb{Z}[q^{\pm1}]$. 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.