The partial (Poincaré) dual with respect to a subset $A$ of edges of a ribbon graph $G$ 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 $\ast$ with Petrie duality $\times$ to give Wilson duality $\ast \times \ast$ and two trialities $\ast \times$ and $\times \ast$. 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-$\ast$ polynomial of $G$, the generating function enumerating partial Poincaré duals by Euler-genus. In this sequel, we introduce the corresponding partial-$\times$, -$\ast \times$, -$\times \ast$, and -$\ast \times \ast$ 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-$\ast \times \ast$ polynomial for a ribbon graph $G$ equals the partial-$\times$ polynomial for $G^{\ast }$. 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.

关于子集的局部对偶（Poincaré）对偶$\mathrm{一种}$ 功能区图的边缘的数量 $G$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$\ast$ 与Petrie对偶 $\times$ 给威尔逊二元性 $\ast \times \ast$和两次审判 $\ast \times$ 和 $\times \ast$。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-$\ast$ 多项式的 $G$，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-$\times$，-$\ast \times$，-$\times \ast$， 和 -$\ast \times \ast$ 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$\ast \times \ast$ 功能区图的多项式 $G$ 等于部分$\times$ 多项式 $G^{\ast }$。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。