European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2021-04-21 , DOI: 10.1016/j.ejc.2021.103329 Jonathan L. Gross , Toufik Mansour , Thomas W. Tucker
The partial (Poincaré) dual with respect to a subset of edges of a ribbon graph was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials. In developing the theory of maps, Wilson and others have composed Poincaré duality with Petrie duality to give Wilson duality and two trialities and . In further expanding the theory, Abrams and Ellis-Monaghan have called the five operators twualities. Part I of this investigation (Gross et al., 2020) introduced as a partial- polynomial of , the generating function enumerating partial Poincaré duals by Euler-genus. In this sequel, we introduce the corresponding partial-, -, -, and - polynomials for their respective twualities. For purposes of computation, we express each partial twuality in terms of the monodromy of permutations of the flags of a map. We analyze how single-edge partial twualities affect the three types (proper, untwisted, twisted) of edges. Various possible properties of partial-twuality polynomials are studied, including interpolation and log-concavity; machine-computed unimodal counterexamples to some log-concavity conjectures from Gross et al. (2020) are given. It is shown that the partial- polynomial for a ribbon graph equals the partial- polynomial for . Formulas or recursions are given for various families of graphs, including ladders and, for Wilson duality, a large subfamily of series–parallel networks. All of these polynomials are shown to be log-concave.
中文翻译:
功能区图的部分对偶,II:部分对偶多项式和单峰计算
关于子集的局部对偶(Poincaré)对偶 功能区图的边缘的数量 由Chmutov结合Jones–Kauffman和Bollobás–Riordan多项式引入。在发展地图理论时,威尔逊和其他人组成了庞加莱对偶 与Petrie对偶 给威尔逊二元性 和两次审判 和 。为了进一步扩展该理论,艾布拉姆斯和埃利斯·莫纳汉(Ellis-Monaghan)称这五名操作员为二人。本次调查的第一部分(建筑等,2020)引入一个partial- 多项式的 ,该生成函数枚举Euler属的部分Poincaré对偶。在这部续集中,我们介绍了相应的partial-,-,-, 和 - 多项式的多项式。为了计算的目的,我们根据映射标志的排列的单价性来表示每个部分二重性。我们分析了单边部分成对扭曲如何影响边缘的三种类型(正确,不扭曲,扭曲)。研究了部分二元多项式的各种可能性质,包括插值和对数凹度;机器计算的单峰反例反驳了Gross等人的一些对数凹度猜想。(2020)。结果表明,部分 功能区图的多项式 等于部分 多项式 。给出了各种图形族的公式或递归,包括梯形图,对于威尔逊对偶性,给出了一个大的系列-并行网络子族。所有这些多项式都显示为对数凹形。