由简单多面体的1-骨架产生的偏序集、非重访路径猜想与偏序集拓扑
Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
AI总结：
本文研究由简单多面体1-骨架经通用线性泛函定向所得格结构，证明共同覆盖元素的并等于最小面的汇，提出无向路径不重访面的猜想及其对单纯形法高效性的意义，并证明开区间序复形同伦等价于球或球面，应用于弱Bruhat序、Tamari格和Cambrian格。
摘要：
给定任意多面体$P$和任意通用线性泛函${\f c}$，通过对每条边$e(u,v)$按${\f c}(u) < {\f c}(v)$从$u$定向到$v$，可从$P$的1-骨架得到一个有向图$G(P,{\f c})$。对于$P$为简单多面体且$G(P,{\f c})$为格$L$的Hasse图的情形，证明了$L$中任意一组共同覆盖同一元素$u$的元素集合$S$的并（join）等于$P$中包含$u$及$S$中所有元素的最小面的汇（sink）。作者猜想对于这样的$G(P,{\f c})$，$G(P,{\f c})$中不存在任何有向路径会重访$P$的任意一个面（facet）。这将意味着对于这样的$P$和${\f c}$，线性规划的单纯形法在所有可能的枢轴规则下都是高效的。该猜想对3-多面体和纺锤体（spindles）已获证明。对于$G(P,{\f c})$为格$L$的Hasse图的简单多面体，证明了$L$中每个开区间的序复形（order complex）同伦等价于一个球或一个球面。文中给出了对弱Bruhat序、Tamari格和Cambrian格的应用。本文以Dominik Preuß的附录作结，该附录证明了当$P$为简单多面体且$G(P,{\f c})$为格的Hasse图时的单调Hirsch猜想。这证实了作者猜想所蕴含的主要推论之一。
英文摘要：
Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice $L$, the join of any collection $S$ of elements which all cover a common element $u$ in $L$ is proven to equal the sink of the smallest face of $P$ containing $u$ and all of the elements of $S$. The author conjectures for such $G(P,{\bf c})$ that no directed path in $G(P,{\bf c})$ ever revisits any facet of $P$. This would imply for such $P$ and ${\bf c}$ that the simplex method for linear programming is efficient under all possible pivot rules. This conjecture is proven for 3-polytopes and for spindles. For simple polytopes in which $G(P,{\bf c})$ is the Hasse diagram of a lattice $L$, the order complex of each open interval in $L$ is proven homotopy equivalent to a ball or a sphere. Applications are given to the weak Bruhat order, the Tamari lattice, and the Cambrian lattices. This paper concludes with an appendix by Dominik Preußproving the monotone Hirsch conjecture for $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice. This confirms one of the main consequences that the author's conjecture would have.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
