标题： 量子态系统计算完美匹配数 显示英文标题

标题： Quantum state systems that count perfect matchings

摘要： 在本文中，我们展示了如何对$n$-色顶点多项式进行范畴化，该多项式基于罗杰·彭罗斯的一个公式，用于计算平面三价图的$3$-边着色数量。 利用拓扑量子场论（TQFT），我们引入了一个量子态系统，以构建一个新的双分次理论，称为双分次$n$-色顶点同调。 该同调的分次欧拉特征是$n$-色顶点多项式。 然后，我们生成一个谱序列，其$E_\infty$页是一个称为过滤的$n$-色顶点同调的过滤理论，并证明它由某些类型的带边图面着色生成。 对于$n=2$，我们证明过滤后的$n$-色顶点同调群由对应完美匹配的面着色生成。 最后，我们引入并解释了当$n \geq 2$时顶点多项式所计数的意义。 这个多项式是一个新的抽象图不变量，可以从 Penrose 的某些公式中推导出来。 显示英文摘要

摘要： In this paper we show how to categorify the $n$-color vertex polynomial, which is based upon one of Roger Penrose's formulas for counting the number of $3$-edge colorings of a planar trivalent graph. Using topological quantum field theory (TQFT), we introduce a quantum state system to build a new bigraded theory called the bigraded $n$-color vertex homology. The graded Euler characteristic of this homology is the $n$-color vertex polynomial. We then produce a spectral sequence whose $E_\infty$-page is a filtered theory called filtered $n$-color vertex homology and show that it is generated by certain types of face colorings of ribbon graphs. For $n=2$, we show that the filtered $n$-color vertex homology is generated by face colorings that correspond to perfect matchings. Finally, we introduce and give meaning to what the vertex polynomial counts when $n \geq 2$. This polynomial is a new abstract graph invariant that can be inferred from certain formulas of Penrose.