Point configurations, phylogenetic trees, and dissimilarity vectors

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.