# The Flexibility and Rigidity of Leaper Frameworks

## TL;DR

This paper proves a conjecture that certain leaper frameworks are rigid on large square grids and flexible on smaller rectangular grids, fully resolving their flexibility and rigidity properties.

## Contribution

It confirms Solymosi and White's conjecture on leaper framework rigidity and characterizes the boundary between flexibility and rigidity on rectangular grids.

## Key findings

- Leaper frameworks are rigid on square grids of size at least 2(p+q)-1.
- Leaper frameworks are flexible on smaller rectangular grids.
- The rigidity and flexibility thresholds are precisely determined for all grid sizes.

## Abstract

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.