Abstract

A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra \(\mathfrak{sl}_2\), but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams.

In the present paper we give a formula for the values of the \(\mathfrak{sl}_2\) weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts.

Our main computational tool is the Chmutov–Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the \(\mathfrak{sl}_2\) weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S. K. Lando on the values of the weight system \(\mathfrak{sl}_2\) at the projections of chord diagrams on the space of primitive elements.

中文翻译：

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完全二分图上$$ \ mathfrak {sl} _2 $$权重系统的值

摘要

加权系统是和弦图上的函数，它满足所谓的四项关系。瓦西里耶夫（Vassiliev）的有限阶结不变量理论用权重系统描述了这些不变量。特别地，存在与彩色琼斯多项式相对应的权重系统。可以根据李代数\（\ mathfrak {sl} _2 \）轻松定义此权重系统，但是从计算的角度来看，此定义过于繁琐，因此该权重系统的值仅在某些情况下才为人所知有限的和弦图类。

在本文中，我们为一类和弦图（其交点图是完全二部图且其中一个部分不超过三个顶点）的弦图给出了\（\ mathfrak {sl} _2 \）权重系统的值的公式。

我们的主要计算工具是Chmutov-Varchenko递归关系。此外，一个部分中不超过三个顶点的完整二部图生成图的Hopf代数的Hopf子代数，并推导沿着这些子代数中可分解元素的子空间投影到基本元素的子空间上的公式。我们为具有此类相交图的和弦图的投影计算\（\ mathfrak {sl} _2 \）权重系统的值。我们的结果证实了由于SK Lando在原始元素空间上的弦图投影上的权重系统\（\ mathfrak {sl} _2 \）的值上的某些猜想。