A tropical approach to rigidity: counting realisations of frameworks

TL;DR

This paper uses tropical geometry to characterize and bound the number of complex solutions for minimally rigid frameworks in the plane, linking geometric rigidity with combinatorial matroid theory.

Contribution

It introduces a tropical geometric method to compute the realisation number of minimally rigid graphs, connecting rigidity to Bergman fans and the Tutte polynomial.

Findings

Provides a combinatorial upper bound on the realisation number.

Shows the bound often improves upon the mixed volume bound.

Establishes a link between rigidity, tropical geometry, and matroid theory.

Abstract

A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points. For generic realisations, the size of the solution set depends only on the underlying graph so long as we allow for complex solutions. We provide a characterisation of the realisation number - that is the cardinality of this complex solution set - of a minimally rigid graph. Our characterisation uses tropical geometry to express the realisation number as an intersection of Bergman fans of the graphic matroid. As a consequence, we derive a combinatorial upper bound on the realisation number involving the Tutte polynomial. Moreover, we provide computational evidence that our upper bound is usually an improvement on the mixed volume bound.