发表机构
Technische Universität Graz; Indiana University Southeast（格拉茨工业大学; 印第安纳大学东南分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出对应框架格的多面体复形framingtope，证明其三种等价描述，构造热带实现并给出多类带框图的显式顶点坐标，复现经典排列多面体等结构。
AI 中文摘要
框架格源自流形多面体的带框（或称DKK）三角剖分的对偶图，为布尔格、Tamari格、弱序格等经典格以及某些Gentle代数的τ-倾斜偏序集提供了统一框架。本文提出framingtope（框架多面体复形），这是一种对应框架格的多面体复形，其边图即为框架格的哈塞图。我们证明framingtope存在三种等价描述：带框三角剖分的内面、覆盖图的两两相容路径集合、框架格的纯区间。我们进一步构造了framingtope的热带实现，将其作为与容许高度函数相关的热带超曲面排列的有界胞腔复形。该构造为平面带框图、多oruga图等大类带框图形提供了显式顶点坐标。在多oruga情形下，这些坐标给出了多重排列上弱序的热带实现；在普通oruga情形下，则可复现经典的排列多面体。
英文摘要
Framing lattices arise from the dual graphs of framed (or DKK) triangulations of flow polytopes and provide a common framework encompassing classical lattices such as the Boolean, Tamari, and weak-order lattices, as well as $τ$-tilting posets of certain gentle algebras. In this paper, we introduce the \emph{framingtope}, a polytopal complex that provides a geometric counterpart to a framing lattice: its edge graph is the Hasse diagram of the framing lattice. We prove that the framingtope admits three equivalent descriptions, in terms of interior faces of the framed triangulation, sets of pairwise coherent routes covering the graph, and pure intervals of the framing lattice. We further construct a tropical realization of the framingtope as the bounded-cell complex of an arrangement of tropical hypersurfaces associated with an admissible height function. This construction yields explicit vertex coordinates for broad classes of framed graphs, including plane framed graphs and multioruga graphs. In the multioruga case, these coordinates give tropical realizations of weak orders on multipermutations and, in the ordinary oruga case, recover the classical permutahedron.
