Tropicalizing binary geometries

Shelby Cox; Igor Makhlin

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arXiv · 2410.13652

Tropicalizing binary geometries

Shelby Cox·Igor Makhlin

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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.

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Shelby Cox, Igor Makhlin. 2024-10-17. Tropicalizing binary geometries. https://arxiv.org/abs/2410.13652

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