分圆牛顿展开与秩一致的整数值牛顿完备化
Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
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中文总结 AI 辅助
本文证明了Chen--Liu--Zhu分圆展开猜想，确定了秩一致展开的自然系数环，构造的牛顿系数可恢复对称染色HOMFLY--PT多项式，其正秩特化有限且可在单位根处求值。
中文摘要 AI 辅助
设$J_r^{SU(n)}(K;q)$为零标架纽结$K$的约化$SU(n)$量子不变量，其由定义表示的第$r$个对称幂染色，且归一化后对未纽结取值为1。对每个固定的$n\ge2$，我们证明Chen--Liu--Zhu分圆展开猜想：存在唯一系数$H_k^{(n)}(K;q)\in\mathbb{Z}[q^{\pm1}]$，使得$J_r^{SU(n)}(K;q)=\sum_{k=0}^{r} \left(\prod_{i=0}^{k-1}\{r-i\}\{r+n+i\}\right) H_k^{(n)}(K;q)$，其中$\{m\}=q^m-q^{-m}$。Beliakova--Gorsky的有限对偶插值公式给出了一个整系数单侧阶乘展开式。在将其约化标量与Habiro--Lê约定匹配后，我们将完备中心限制为单行染色。完备的Harish--Chandra反射在变量$z$中产生反演对称性，通过$X=z+z^{-1}$的整系数下降将单侧展开转化为双侧牛顿基。分圆局部插值和唯一分解整环（UFD）分母消去论证证明了牛顿系数的Laurent整性。我们还确定了秩一致展开的自然系数环：对每个零标架纽结，存在唯一的Laurent微分系数$G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$，其对应的牛顿系数是$\mathbb{Q}(q)$上$A$的Laurent多项式，且在每个几何节点$A=q^n$（$n\ge2$）处的取值属于$\mathbb{Z}[q^{\pm1}]$。这些系数定义了一个两变量牛顿逆极限元素，其正秩特化可恢复所有对称染色的HOMFLY--PT多项式。该完备化是在牛顿核中进行，而非在单位根处逐系数完成。在固定正秩和染色后，该级数有限且可在单位根处求值。
英文摘要
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--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 $G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$. 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.
