Betti numbers of cut ideals of trees

TL;DR

This paper investigates the algebraic properties of cut ideals of trees, providing upper bounds for their Betti numbers through combinatorial and topological methods, with formulas depending on the number of vertices.

Contribution

It introduces new upper bounds for Betti numbers of cut ideals of trees using topological combinatorics, linking algebraic invariants to graph enumeration.

Findings

Derived simple formulas for Betti number bounds based on vertex count

Connected Betti number bounds to enumeration of induced subgraphs

Applied topological combinatorics to algebraic statistics problems

Abstract

Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from topological combinatorics, we obtain upper bounds for the Betti numbers of this type of ideals. These take the form of simple formulas on the number of vertices, which arise from the enumeration of induced subgraphs of certain incomparability graphs associated to the edge sets of trees.