嵌入图的扭曲对偶
Twisted duality for embedded graphs
摘要：
我们考虑嵌入图（或等价的带状图）的边缘上的两个操作：给边缘半扭曲和关于边缘取部分对偶。这两个操作产生一个作用S_3^{|E（G）|}，功能区组，在G.带状群在嵌入图上的作用推广了对偶、部分对偶和Petrie对偶的概念。我们证明了这种带状群作用给出了对偶的一个完整刻画，即如果G是具有中间图G_m的任意胞腔嵌入图，则G在群作用下的轨道恰好是所有具有与G_m同构的中间图的图（作为抽象图）的集合.我们提供的特殊套扭曲的，如部分的扭曲，嵌入图的媒体图方面的特征，我们展示了如何不同种类的图同构引起这些不同的概念的对偶。带状群作用，然后导致更深入地了解的性质，和之间的关系，各种图形多项式通过广义的过渡多项式，自然地相互作用的带状群作用。
英文摘要：
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|}, the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality and Petrie duality. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph with medial graph G_m, then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to G_m. We provide characterizations of special sets of twisted duals, such as the partial duals, of embedded graphs in terms of medial graphs and we show how different kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action.
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