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Rational tensegrities through the lens of toric geometry

Fatemeh MohammadiXian Wu
Dec 2022
摘要
A classical tensegrity model consists of an embedded graph in a vector spacewith rigid bars representing edges, and an assignment of a stress to every edgesuch that at every vertex of the graph the stresses sum up to zero. Thetensegrity frameworks have been recently extended from the two dimensionalgraph case to the multidimensional setting. We study the multidimensionaltensegrities using tools from toric geometry. For a given rational tensegrityframework $\mathcal{F}$, we construct a glued toric surface $X_\mathcal{F}$. Weshow that the abelian group of tensegrities on $\mathcal{F}$ is isomorphic to asubgroup of the Chow group $A^1(X_\mathcal{F};\QQ)$. In the case of planarframeworks, we show how to explicitly carry out the computation of tensegritiesvia classical tools in toric geometry.
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