A tropical approach to rigidity: counting realisations of frameworks
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arXiv
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| Auteurs principaux: | , , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929715531481088 |
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| author | Clarke, Oliver Dewar, Sean Tripp, Daniel Green Maxwell, James Nixon, Anthony Ren, Yue Smith, Ben |
| author_facet | Clarke, Oliver Dewar, Sean Tripp, Daniel Green Maxwell, James Nixon, Anthony Ren, Yue Smith, Ben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A tropical approach to rigidity: counting realisations of frameworks Clarke, Oliver Dewar, Sean Tripp, Daniel Green Maxwell, James Nixon, Anthony Ren, Yue Smith, Ben Combinatorics Algebraic Geometry Metric Geometry 52C25 (Primary) 14T90, 14T15 (Secondary) 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. |
| title | A tropical approach to rigidity: counting realisations of frameworks |
| topic | Combinatorics Algebraic Geometry Metric Geometry 52C25 (Primary) 14T90, 14T15 (Secondary) |
| url | https://arxiv.org/abs/2502.10255 |