In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of $SU(N)$ Chern-Simons Wilson loops, while the limit is $N\to 0$. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ-functions. This result is an extension of what we did in [1], where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton τ-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton τ-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants.

在本文中，我们研究了有色HOMFLY多项式在特定极限下的群理论结构。群结构出现在$\mathrm{小号}ü（ñ）$ Chern-Simons Wilson循环，而极限是 $ñ\to 0$。论文的结果是双重的。首先，我们解释Kadomsev-Petviashvily（KP）τ-函数的出现。该结果是我们在[1]中所做的扩展，其中在KP方程和群因子之间建立了符号对应关系。在本文中，我们证明了彩色亚历山大多项式的可积性是由于它与孤子τ函数的关系。主要是，彩色亚历山大多项式嵌入在孤子τ上的KP生成函数的作用中-功能。其次，我们使用这种对应关系，以杨氏图的形式提供了对群因子的相当简单的组合描述，否则以和弦图的形式描述，其中没有简单的描述。这是第一步，提供了对Wilson循环的群理论数据的明确描述，这将有效地将它们简化为纯粹的拓扑量，主要是Vassiliev不变量的集合。