H-Representation of the Kimura-3 Polytope

TL;DR

This paper derives a half-space representation for the Kimura-3 polytope, a key geometric object in phylogenetics, facilitating better understanding of the associated toric variety.

Contribution

It provides the first explicit half-space description of the Kimura-3 polytope for the m-claw tree, advancing geometric understanding in phylogenetics.

Findings

Derived the half-space representation for the Kimura-3 polytope

Enabled new geometric analyses of the Kimura-3 model

Facilitated computations related to the associated toric variety

Abstract

Given a group-based Markov model on a tree, one can compute the vertex representation of a polytope which describes the associated toric variety. The half-space representation, however, is not easily computable. In the case of $\mathbb{Z}_2$ or $\mathbb{Z}_2\times\mathbb{Z}_2$, these polytopes have applications in the field of phylogenetics. We provide a half-space representation for the $m$-claw tree where $G=\mathbb{Z}_2\times\mathbb{Z}_2$, which corresponds to the Kimura-3 model of evolution.