arXiv 2602.118922026-02math.CO

关于平面刚性模体

On plane rigidity matroids

Charles University（查理大学）

Mykhaylo Tyomkyn

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AI总结：

本文研究了二维刚性模体族的性质，证明了R和H是唯一与代数曲线相关的2-刚性族，并揭示了立方图的定向特性。

摘要：

我们证明了关于与二维刚性相关的模体及其模体族的几个结果。特别是，我们确立了通用刚性模体族R和Kalai的超连通性模体族H的新性质。我们证明R是唯一的2-刚性模体族，其中K_{3,3}不是回路。作为此结果和Bolker-Roth定理的几何推论，得出H和R是唯一与实数域R^2中代数曲线相关的2-刚性族。Bernstein利用热带几何来刻画H-独立图作为那些允许边排序无有向环和交替闭合轨的图。我们为充分性方向提供了组合证明，并将Bernstein定理推广到正特征领域。由此得出n个通用点在n-2维的楔幂模体不依赖于域特征。我们的证明方法允许识别许多在每个2-刚性族中独立的图。特别是，我们证明了所有连通立方图，除了K_4和K_{3,3}外都满足这一条件。这给出了立方图的完整分类，并以强形式回答了Kalai的问题。作为推论，我们获得了一个新的立方图性质：除了K_4和K_{3,3}外，每个连通立方图都有一个无有向环和交替环的定向。等价地说，它可以以特殊的方式分割成两个森林。

英文摘要：

We establish new properties of matroids and matroidal families associated with rigidity in dimension $2$, including the generic rigidity matroid family $\mathcal{R}$ and Kalai's hyperconnectivity matroid family $\mathcal{H}$. Answering a question of Kalai in a strong form, we show that all connected cubic graphs, with exceptions of $K_4$ and $K_{3,3}$, are independent in every $2$-rigidity family. We also prove that $\mathcal{R}$ is the unique matroidal $2$-rigidity family in which $K_{3,3}$ is not a circuit. As a geometric corollary of this result and the Bolker-Roth theorem, it follows that $\mathcal{H}$ and $\mathcal{R}$ are the only $2$-rigidity families associated with algebraic curves in $\mathbb{R}^2$. Bernstein used tropical geometry to characterize $\mathcal{H}$-independent graphs as those admitting an edge-ordering without directed cycles and alternating closed trails. We provide a combinatorial proof of the sufficiency direction, extending Bernstein's theorem to positive characteristic. It follows that the wedge power matroid of $n$ generic points in dimension $n-2$ does not depend on the field characteristic. As a corollary, we obtain a new property of cubic graphs: every connected cubic graph except $K_4$ and $K_{3,3}$ has an orientation without directed and alternating cycles. The current proof of this purely graph theoretic statement relies on tropical geometry and matroid theory.

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