On the Genus of One Degree of Freedom Planar Linkages via Tropical Geometry

TL;DR

This paper investigates the genus of configuration spaces of certain planar linkages using tropical geometry, providing an algorithm and Python implementation to compute the genus of these one-dimensional complex curves.

Contribution

It introduces a novel algorithm based on tropical geometry to compute the genus of configuration spaces of planar linkages, with an implementation and examples.

Findings

Algorithm successfully computes genus for various examples.

Configuration spaces are generically smooth complex curves.

Provides a practical tool for analyzing linkage configuration spaces.

Abstract

This paper focuses on studying the configuration spaces of graphs realised in $\mathbb C^2$, such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a smooth complex curve, and can be seen as a Riemann surface. The property of interest in this paper is the genus of this curve. Using tropical geometry, we give an algorithm to compute this genus. We provide an implementation in Python and give various examples.