arXiv subjects

Rebecca Monks

Publications and source records attributed to Rebecca Monks.

2 recordsLinked to original sources

arXiv2026

Rigidity on compact surfaces through hyperbolic symmetries

Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The remaining case of compact surfaces of genus at least two has remained open. Using the hyperbolic geometry of their universal covers, we develop a theory of infinitesimal rigidity for frameworks on compact surfaces of genus at least two. By the uniformisation theorem, every such surface is a quotient of the hyperbolic plane by a surface group, allowing frameworks on the surface to be represented as infinite symmetric frameworks in the hyperbolic plane. Encoding the symmetry through gain graphs, we prove that infinitesimal rigidity is determined entirely by finite combinatorial data. Specifically, a framework is generically rigid if and only if its associated gain graph contains a spanning (2,3,1,0)-gain tight subgraph. This yields the first combinatorial characterisation of generic rigidity for frameworks on compact surfaces of genus at least two.

math.DG↗

arXiv2025

5-regular graphs and the 3-dimensional rigidity matroid

A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity matroid of $G$. The combinatorial nature of this matroid is well understood when $d=1,2$ but open when $d\geq 3$. Jackson and Jord\'an 2005 characterised independence in this matroid for connected graphs with minimum degree at most $d+1$ and maximum degree at most $d+2$. Their characterisation is known to be false for $(d+2)$-regular graphs when $d\geq 4$ but when $d=3$ it remained open. Indeed they conjectured that their characterisation extends to 5-regular graphs when $d=3$. The purpose of this article is to prove their conjecture. That is, we prove that every 5-regular graph that has at most $3n-6$ edges in any subgraph on $n\geq 3$ vertices is independent in the generic 3-dimensional rigidity matroid.

math.CO↗