The polytopal structure of the tight-span of a totally split-decomposable metric

TL;DR

This paper characterizes the polytopal structure of the tight-span of totally split-decomposable metrics, extending previous results and enabling direct determination of its structure, with potential applications in phylogenetics.

Contribution

It provides a canonical decomposition of the tight-span into polytopal complexes directly derived from the metric, generalizing earlier findings about cell structures.

Findings

The tight-span can be decomposed into polytopal complexes.

The polytopal structure can be directly determined from the metric.

This understanding may improve phylogenetic inference techniques.

Abstract

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split decomposable (finite) metric. Totally split-decomposable metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-spanof a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric in a very direct way.We anticipate that our improved understanding of this structure may ultimately lead to improved techniques for phylogenetic inference.