树演算的KNTZ技巧与微分展开式的结构
KNTZ trick from arborescent calculus and the structure of differential expansion
摘要：
最近提出的KNTZ技巧完成了对所有矩形表示的排他Racah矩阵$\bar S$和$S$的持久搜索，并且在非矩形情况下也有帮助的潜力。这是最后一个缺乏洞察力的结构的微分展开的（矩形）有色结多项式的扭曲结-和由此产生的成功是一个壮观的成就，现代纽结理论在经典领域的表示理论，这最初被认为是一个工具，结演算，但似乎是其直接受益者。在本文中，我们解释说，KNTZ的anastomy实际上是一个建议，转换成三角形形式${\cal B}$的树木进化矩阵$\bar S\bar T^2\bar S$和演示如何工作，什么是从这个角度来看，旧的难题和奇迹的微分展开式的形式。主要的新的完整的结果是猜想的三角矩阵${\cal B}$在非矩形表示$[3,1]$的情况下。本文并没有简化任何计算，而是强调了剩下的问题，为了{ \it证明}事情确实有效，需要克服这些问题。我们相信，这一讨论也是有用的进一步的步骤对非矩形的情况下，相关的搜索规范不变的树顶点。作为一个例子，我们制定了一个令人困惑的，仍然实验支持的猜想，研究扭曲结只足以描述的形状的微分膨胀的所有节点。
英文摘要：
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.
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