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ITEP（核子物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发现当$q^2$为本原单位根时，HOMFLY-PT多项式具有与纽结无关的分解，因子为$A^m$处的特殊多项式，并通过Murnaghan-Nakayama规则和Adams运算证明，解释了长期存在的分解之谜。
AI 中文摘要
HOMFLY-PT多项式描述了Chern-Simons理论中Wilson平均对两个独立参数$q$和$A$的任意复数值的解析延拓。它们还依赖于纽结和规范群的表示。然而，当$q^2$是$m$次本原单位根时，例如$q=\pm e^{i\pi/m}$，这些多项式表现出一种普遍的（与纽结无关的）分解：$H_R$成为Young图$R$的$m$-核对应的多项式与每个$m$-带对应的同一因子$H_{[m]}$的乘积。在我们计算的所有例子中，该因子恰为$A^m$处的特殊多项式。当$q=\pm1$时，分解源于缆化HOMFLY的奇异性，该奇异性对所有表示相同。当$m>1$时，相关奇异性出现在投影子中，其幸存部分为辫子$X_m$的有限和——即缆内$m$次幂和，可自由穿过其他股。证明中我们使用了Murnaghan-Nakayama规则和Adams运算。这一隐藏结构的发现解释了旧的分解之谜。
英文摘要
HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\pm e^{iπ/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ -- the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.
Comments24 pages
