Tropicalizing binary geometries

TL;DR

This paper explores the tropicalization of type A and C cluster configuration spaces, linking them to phylogenetic trees and polyhedral subfans, and proposes an extension from type A to type C.

Contribution

It provides a concise overview of tropicalizing type A cluster spaces and introduces a conjectural extension to type C involving cyclohedra and symmetric phylogenetic trees.

Findings

Tropicalization of $ ext{M}_{0,n}$ corresponds to phylogenetic trees.

Signed tropicalizations form dual-associahedron and dual-cyclohedron subfans.

Proposes a conjectural realization of type C tropicalization as axially symmetric phylogenetic trees.

Abstract

The type A cluster configuration space, commonly known as $\mathcal M_{0,n}$, is the very affine part of the binary geometry associated with the associahedron. The tropicalization of $\mathcal M_{0,n}$ can be realized as the space of phylogenetic trees and its signed tropicalizations as the dual-associahedron subfans. We give a concise overview of this construction and propose an extension to type C. The type C cluster configuration space $\mathcal M_{\mathrm C_l}$ arises from the binary geometry associated with the cyclohedron. We define a space of axially symmetric phylogenetic trees containing many dual-associahedron and dual-cyclohedron subfans. We conjecturally realize the tropicalization of $\mathcal M_{\mathrm C_l}$ as the defined space and its signed tropicalizations as the aforementioned subfans.