发表机构
Westlake University; Peking University（西湖大学; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了矩形分圆展开猜想和显式双扭转公式，构造了系数唯一的双牛顿展开，并利用表示对称性和积分插值，为任意零框架纽结的彩色HOMFLY-PT多项式提供了统一框架。
AI 中文摘要
我们证明了Kameyama、Nawata、Tao和Zhang关于任意零框架纽结的约化彩色HOMFLY-PT多项式的矩形分圆展开猜想。我们还证明了对于每个矩形颜色和所有非零整数扭转参数对，他们的显式双扭转公式。对于每个纽结，我们构造了一个系数在$\mathbb Z[A^{\pm1},q^{\pm1}]$中的双牛顿展开。这些系数是唯一确定的，并且独立于矩形颜色的宽度和高度。该构造使用了有限秩表示对称性和积分插值。双扭转公式由两个四点Casimir运算符的同时外幂比较和已知的单行公式得出。有限过渡公式将双扭转纽结的双牛顿系数表示为规定的扭转系数。
英文摘要
We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in $\mathbb Z[A^{\pm1},q^{\pm1}]$. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.
Comments32 pages, 1 figure
