Generalized metric tree arrangements and Dressians

TL;DR

This paper introduces generalized metric tree arrangements that parametrize points in Dressians $Dr(k,n)$, extending previous results and analyzing cone adjacency in the positive Dressian $Dr_{>0}(3,n)$ using generalized Whitehead moves.

Contribution

It extends metric tree arrangements to general Dressians $Dr(k,n)$ and studies cone adjacency in $Dr_{>0}(3,n)$ with new combinatorial moves.

Findings

Generalized metric tree arrangements parametrize points in $Dr(k,n)$.

Explicit examples of generalized metric tree arrangements are provided.

A new condition for cone adjacency in $Dr_{>0}(3,n)$ using generalized Whitehead moves.

Abstract

Metric trees and metric tree arrangements index cones in the polyhedral fan structure in the Dressian $Dr(2,n)$ and $Dr(3,n)$ respectively. We introduce the notion of generalized metric tree arrangements which parameterize points in $Dr(k,n)$ and extend previously known results to $Dr(k,n)$ along with providing explicit examples of these generalized metric tree arrangements. We study the adjacency of cones in the positive Dressian $Dr_{>0}(3, n)$ and introduce generalized Whitehead moves which provide a condition for adjacency of maximal cones in $Dr_{>0}(3,n)$ in terms of the associated metric tree arrangements.