对称表示中二分HOMFLY多项式的平面分解
Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
AI总结：
本文将二分纽结图HOMFLY多项式的平面分解推广到（反）对称着色情形，通过投影恢复平面性，超越树状演算，并给出计算实例，揭示二分演化与反平行通道中R-矩阵特征值平方的分解。
摘要：
我们将最近发现的二分纽结/链环图HOMFLY多项式的平面分解（Kauffman括号）推广到（反）对称着色的HOMFLY多项式。缆化会破坏平面性，但在投影到（反）对称表示后平面性得以恢复。这使得我们能够超越树状演算，而迄今为止树状演算产生了着色多项式的大部分结果。技术细节包括投影子的组合，这些可以严格处理，无需任何猜测——这对其他迄今缺乏可靠量子化的考虑也可能有用。我们明确给出了使用我们的平面技术计算对称表示中HOMFLY多项式的简单例子。这些例子揭示了我们所称的二分演化以及反平行通道中$\mathcal{R}$-矩阵特征值平方的二分分解。
英文摘要：
We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows to go beyond arborescent calculus, which so far produced the majority of results for colored polynomials. Technicalities include combinations of projectors, and these can be handled rigorously, without any guess-work -- what can be also useful for other considerations, where reliable quantization was so far unavailable. We explicitly provide simple examples of calculation of the HOMFLY polynomials in symmetric representations with the use of our planar technique. These examples reveal what we call the bipartite evolution and the bipartite decomposition of squares of $\mathcal{R}$-matrices eigenvalues in the antiparallel channel.
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