Summary
Some old ideas in math thought every special shape called an oriented matroid had a simple kind of face, but this paper shows a counterexample where none exist. It also shows that certain transformations between these shapes can't always connect all shapes, disproving another guess from decades ago. The authors built specific examples proving these old beliefs wrong.
What this means in practice
- •For combinatorial system designers: Identify limitations in mutation-based transformations when modeling combinatorial structures using oriented matroids of certain sizes and ranks.
- •For mathematical software developers: Avoid assuming mutation connectivity in algorithms handling oriented matroids, improving the correctness of combinatorial computation tools.
A theory result. No direct application yet.
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.