Neither simpliciality nor mutation connectivity: conjectures of Las Vergnas and Cordovil-Las Vergnas fail

2026-09-14

AI Summary

本文通过构建一个特定的定向拟阵，反驳了Las Vergnas单形猜想，并证明了对于无限多秩和基集大小，定向拟阵的变异图是不连通的，从而推翻了Cordovil-Las Vergnas猜想。

Abstract

We construct a simple rank-$7$ oriented matroid on $24$ elements with no simplicial tope, disproving the Las Vergnas simplex conjecture from 1980. We then show that the mutation graph on uniform oriented matroids is disconnected for infinitely many values of rank $r$ and ground-set size $n$, disproving the Cordovil--Las Vergnas conjecture from 1988.

Problem

• oriented matroid
• simplicial tope
• mutation graph

Innovation

• oriented matroid
• simplicial tope
• mutation graph