论文标题
图形和嵌入式图的偏斜特征多项式
Skew characteristic polynomial of graphs and embedded graphs
论文作者
论文摘要
我们介绍了一个新的单变量多项式不变的图，我们称之为偏斜的特性多项式。对于定向的简单图，这只是其反对称邻接矩阵的特征多项式。对于非方向的简单图，定义是不同的，但是对于某些类别的图形（即，对于和弦图的相交图），如果我们将这样的图赋予了和弦图引起的方向，则它给出了相同的答案。我们证明，这种不变的人满足了Vassiliev的$ 4 $ - 期关系，因此确定有限的结。我们研究了多项式相对于HOPF代数结构在图形空间上的行为，并表明它对此HOPF代数中的任何原始元素都具有恒定值。我们还提供了偏斜特性多项式到嵌入式图和甲状腺的两变量扩展。扩展多项式的$ 4 $ - 期关系证明它确定了多组件链接的有限类型不变。
We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The $4$-term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.
