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ITEP（ITEP）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从微分展开出发，将扭转结和反平行双辫结在矩形表示下的HOMFLY-PT多项式表示为行列式，为结多项式扩展至KP/Toda τ-函数提供新起点。
AI 中文摘要
从已知的微分展开出发，我们将矩形表示[r^s]中的扭转结和反平行双辫结的HOMFLY-PT多项式表示为r×r和s×s矩阵的行列式。我们首先对八字结给出一个初等推导，然后重新求和扭转结和双辫结演化的KNTZ公式。所得矩阵由单钩演化系数和显式的表示依赖权重构成。这些公式为研究结多项式到KP/Toda τ-函数的可能扩展提供了起点。扩展到非矩形表示和更一般的结族仍是未解决的问题。
英文摘要
Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(τ\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.
Comments18 pages
