On the genus of one degree of freedom planar linkages via tropical geometry

This paper focuses on studying the configuration spaces of graphs realised in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb C^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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.