Excluded checkerboard colourable ribbon graph minors

Checkerboard colourable twisted duals

In this note we show that any embedded graph has a checkerboard colourable twisted dual and any Eulerian embedded graph has a checkerboard colourable partial Petrial, answering questions posed by Ellis-Monaghan and Moffatt. The proofs are based on orientations of their medial graphs and orientations of boundary components of their corresponding ribbon graphs. The arrow presentations of ribbon graphs are also used. We also obtain two related results.

Characterization of regular checkerboard colourable twisted duals of ribbon graphs

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. The twisted dual is a further generalization by combining the partial Petrial. Given a ribbon graph $G$, in this paper, we first characterize regular partial duals of the ribbon graph $G$ by using spanning quasi-tree and its related shorter marking arrow sequence set. Then we chara…

On the ribbon graphs of links in real projective space

Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to each other. We also find moves that relate the diagrams of links in real projective space that give rise to (all-A) ribbon graphs with exactly one vertex.

嵌入图的Penrose多项式

A Penrose polynomial for embedded graphs

我们扩展彭罗斯多项式，原来只定义为平面图，图形嵌入在任意表面。考虑嵌入图的Penrose多项式，得到了嵌入图的Penrose多项式在平面图中不能实现的新的恒等式和关系式.特别是，通过利用与过渡多项式和带状群作用的连接，我们发现了一个删除收缩型的彭罗斯多项式的关系。我们与彭罗斯多项式的定向棋盘列图的电路分区多项式的中间图，并使用此找到新的组合解释的彭罗斯多项式。我们还证明了平面图G的Penrose多项式可以表示为G的扭曲图的色多项式之和。这使我们能够得到一个新的四色定理的重新表述。

曲面上的图与Khovanov同调

Graphs on surfaces and Khovanov homology

定向带状图（dessins d'enfant）是嵌入在定向曲面中的图。带状图的拟树是一个只有一个面的生成子图，它可用有序弦图来描述。我们表明，对于任何链接图$L$，有一个相关的带状图，其准树对应双射的棋盘着色$L$得到的图的生成树。这种对应关系保留了用于Khovanov同调生成树模型的二阶化，其欧拉示性数是$L$的琼斯多项式。因此，Khovanov同调可以用带状图表示，生成元由有序弦图给出。

虚杆的Kauffman括号和Bollobás-Riordan多项式

The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial

我们表明，Kauffman括号$[L]$的棋盘可着色虚拟链接$L$是一个评估的Bollobás-Riordan多项式$R_{G_L}$的带状图与$L$。这个结果推广了Thistlethwaite关于可平面图的Kauffman括号与Tutte多项式的著名定理。