The recently suggested KNTZ trick completed the lasting search for exclusive Racah matrices $\bar S$ and $S$ for all rectangular representations and has a potential to help in the non-rectangular case as well. This was the last lacking insight about the structure of differential expansion of (rectangularly-)colored knot polynomials for twist knots -- and the resulting success is a spectacular achievement of modern knot theory in a classical field of representation theory, which was originally thought to be a tool for knot calculus but instead appeared to be its direct beneficiary. In this paper we explain that the KNTZ ansatz is actually a suggestion to convert the arborescent evolution matrix $\bar S\bar T^2\bar S$ into triangular form ${\cal B}$ and demonstrate how this works and what is the form of the old puzzles and miracles of the differential expansions from this perspective. The main new fully result is the conjecture for the triangular matrix ${\cal B}$ in the case of non-rectangular representation $[3,1]$. This paper does not simplify any calculations, but highlights the remaining problems, which one needs to overcome in order to {\it prove} that things really work. We believe that this discussion is also useful for further steps towards non-rectangular case and the related search of the gauge-invariant arborescent vertices. As an example we formulate a puzzling, still experimentally supported conjecture, that the study of twist knots only is sufficient to describe the shape of the differential expansion for all knots.

中文翻译：

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树状微积分的 KNTZ 技巧和微分展开的结构

最近建议的 KNTZ 技巧完成了对所有矩形表示的独家 Racah 矩阵 $\bar S$ 和 $S$ 的持久搜索，并且也有可能在非矩形情况下提供帮助。这是关于扭结的（矩形）彩色结多项式的微分展开结构的最后一次缺乏洞察力——由此产生的成功是现代结理论在经典表示理论领域的一项壮观成就，最初被认为是成为结微积分的工具，但似乎是它的直接受益者。在本文中，我们解释了 KNTZ ansatz 实际上是将树状进化矩阵 $\bar S\bar T^2\bar S$ 转换为三角形形式 ${\cal B}$ 的建议，并演示了它是如何工作的以及什么是从这个角度来看，差异扩张的古老谜题和奇迹的形式。主要的新结果是在非矩形表示 $[3,1]$ 的情况下对三角矩阵 ${\cal B}$ 的猜想。这篇论文没有简化任何计算，而是强调了剩下的问题，需要克服这些问题才能{\it证明}事情真的有效。我们相信，这个讨论对于非矩形情况的进一步步骤和规范不变树状顶点的相关搜索也很有用。作为一个例子，我们提出了一个令人费解的、仍然得到实验支持的猜想，