Point configurations, phylogenetic trees, and dissimilarity vectors

Abstract

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

Keywords: Grassmannian; dissimilarity vector; phylogenetic tree; rational normal curve; tropical geometry.

Fig. 1.

The graph $G$ defining a 10-dimensional cone $\sigma$ in the space of four-dissimilarity vectors.

Fig. 2.

The three graphs whose corresponding 11-dimensional cones ${\tau }_1,{\tau }_2,{\tau }_3$ meet along the common face $\sigma$.