曲面图局部不变量的平面图
Planar diagrams for local invariants of graphs in surfaces
摘要：
为了将量子拓扑方法应用于非平面图，我们定义了一个平面图范畴，它描述了图嵌入曲面的局部拓扑。这些\emph{virtual graphs}是带状图的分类解释。我们描述了一个扩展的流多项式的虚拟图，$S$-多项式，并制定了非三次图的$\mathfrak{sl}(N)$彭罗斯多项式，给出收缩删除关系。利用$S$-多项式定义了Yamada多项式在虚拟空间图上的推广，并由此得到了虚拟空间图非经典的一个充分条件.我们猜想整数平方上的$S$-多项式的局部关系的存在性。
英文摘要：
In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $\mathfrak{sl}(N)$ Penrose polynomial for non-cubic graphs, giving contraction-deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.
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