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

Josef Schicho, Ayush Kumar Tewari, Audie Warren

TL;DR

The paper develops a tropical-geometry framework to compute the genus of the configuration space of 1-dof planar linkages realized in ${\mathbb C}^2$. By tropicalizing the generic fibre of the edge-length map, it reduces the problem to the intersection of two tropical varieties $X$ and $Y$ and proves that, when tropically smooth, the tropical genus matches the algebraic genus of each irreducible component. An explicit, probabilistic algorithm traverses the tropical curve to count bounded edges and vertices, enabling genus computation with a Python implementation and illustrative examples. The study connects rigidity theory (via Laman numbers) with tropical geometry, providing concrete results for graphs like the four-cycle and the family $K_{2,m}$ (and generalizations $O_{l,r}$), including closed-form genus formulas and RH-based recurrences. This work offers a practical method to determine genus in 1-dof linkage spaces and reveals structural patterns, such as the observed odd-genus phenomenon among computed cases.

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.

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

TL;DR

The paper develops a tropical-geometry framework to compute the genus of the configuration space of 1-dof planar linkages realized in . By tropicalizing the generic fibre of the edge-length map, it reduces the problem to the intersection of two tropical varieties and and proves that, when tropically smooth, the tropical genus matches the algebraic genus of each irreducible component. An explicit, probabilistic algorithm traverses the tropical curve to count bounded edges and vertices, enabling genus computation with a Python implementation and illustrative examples. The study connects rigidity theory (via Laman numbers) with tropical geometry, providing concrete results for graphs like the four-cycle and the family (and generalizations ), including closed-form genus formulas and RH-based recurrences. This work offers a practical method to determine genus in 1-dof linkage spaces and reveals structural patterns, such as the observed odd-genus phenomenon among computed cases.

Abstract

This paper focuses on studying the configuration spaces of graphs realised in , 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.
Paper Structure (14 sections, 12 theorems, 26 equations, 9 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 12 theorems, 26 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. The Precise Statement of the Problem, and a Theorem on Tropical Curves
  2. The Precise Statement of the Problem
  3. A Theorem on Tropical Curves
  4. The Tropical Generic Fiber
  5. Tropicalization
  6. Tropical Smoothness
  7. Algorithm
  8. Starting Point / Weight Generation
  9. Direction Calculation
  10. The Genus of a Trace
  11. Examples
  12. The graph C4
  13. The Bipartite Graphs K2m
  14. Further Examples

Key Result

Theorem 1

Let $\mathcal{C}$ be a smooth projective curve of genus $g$, such that $T:=\text{Trop}(\mathcal{C})$ is smooth. Then the tropical genus of $T$ is equal to $g$.

Figures (9)

  • Figure 1: A elliptic curve $C$
  • Figure 2: Tropical elliptic curve $\text{Trop}(C)$
  • Figure 4: The curve $x^3 + y^3 + 1 = 0$
  • Figure 5: The corresponding tropical curve
  • Figure 6: Three traces of the same graph with six vertices. The genus of the configuration curves and the traces is equal to 17.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 1: Smooth Tropical Curve
  • Remark 1
  • Example 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • ...and 17 more