5 papers
The number of realisations of a random graph
Sean Dewar, Anthony Nixon, Ben Smith
Determining the number of realisations, up to isometries, of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove t…
The -dimensional realisation number of a rigid graph
Sean Dewar, Anthony Nixon, Ben Smith
Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide tw…
Counting fibres of the Hadamard product using Bergman fans
Oliver Clarke, Sean Dewar, Matteo Gallet +3
We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associa…
A tropical approach to rigidity: counting realisations of frameworks
Oliver Clarke, Sean Dewar, Daniel Green Tripp +4
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 f…
Geometry of tropical extensions of hyperfields
James Maxwell, Ben Smith
We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as…