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Point configurations, phylogenetic trees, and dissimilarity vectors.

Alessio Caminata¹, Noah Giansiracusa², Han-Bom Moon³

¹Dipartimento di Matematica, Università di Genova, 16146 Genova, Italy.

Proceedings of the National Academy of Sciences of the United States of America

|March 16, 2021

Summary

Dissimilarity vectors from phylogenetic trees do not form a tropical subvariety. However, weighted dissimilarity vectors do, offering a new geometric interpretation and characterization.

Keywords:

Grassmannian dissimilarity vector phylogenetic tree rational normal curve tropical geometry

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Area of Science:

Computational Biology
Algebraic Geometry
Tropical Geometry

Background:

Pachter and Speyer introduced higher dissimilarity maps for phylogenetic trees in 2004.
Previous work confirmed dissimilarity vectors lie on the tropical Grassmannian.

Purpose of the Study:

To determine if dissimilarity vectors form a tropical subvariety of the tropical Grassmannian.
To explore weighted dissimilarity maps and their geometric properties.

Main Methods:

Investigating the tropical balancing condition for dissimilarity vectors.
Developing and analyzing weighted dissimilarity maps.
Utilizing geometric interpretations involving rational normal curves.
Constructing a finite tropical basis.

Main Results:

The tropical balancing condition fails, indicating dissimilarity vectors do not form a tropical subvariety.
Weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian.
A geometric interpretation using configurations of points on rational normal curves was established.
A finite tropical basis for weighted dissimilarity vectors was constructed.

Conclusions:

The set of dissimilarity vectors does not form a tropical subvariety.
Weighted dissimilarity vectors precisely form a tropical subvariety as envisioned by Pachter and Speyer.
The study provides a novel geometric framework and explicit characterization for weighted dissimilarity vectors.