The Flexibility and Rigidity of Leaper Frameworks

A leaper framework is a bar-and-joint framework whose joints are integer points forming a rectangular grid and whose bars correspond to all moves of a given leaper within that grid. We study the flexibility and rigidity of leaper frameworks. Let \(p\) and \(q\) be positive integers such that the \((p, q)\)-leaper \(L\) is free. J\'{o}zsef Solymosi and Ethan White conjectured in 2018 that the leaper framework of \(L\) on the square grid of side \(2(p + q) - 1\), and so on all larger grids, is rigid. We prove this conjecture. We also prove that Solymosi and White's conjecture is, in a sense, sharp. Namely, the leaper framework of \(L\) on the rectangular grid of sides \(2(p + q) - 2\) and \(2(p + q) - 1\), and so on all smaller grids (except for, trivially, the \(1 \times 1\) grid), is flexible. In particular, we completely resolve the flexibility and rigidity question for leaper frameworks on square grids. We establish a number of related results as well.