Betti Numbers of Cut Ideals of Trees

Abstract

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 combinatorial topology, 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. 2000 Mathematics Subject Classifications: Primary 05C05, 13D02; Secondary 05C25, 13F55

Retracts and algebraic properties of cut algebras

“…The "only if" part of all conjectures had been verified in [28]. Recently these conjectures as well as some related questions were studied by many authors, e.g., in [3,6,12,20,22,23,25,26]. Note that Engström gave an affirmative answer to the first conjecture in [12].…”

Cycle algebras and polytopes of matroids

“…In the particular case of cut polytopes, the aforementioned toric algebras and their defining ideals, which were studied first in [34], are called cut algebras and cut ideals, respectively. For further studies around cut algebras and ideals, see, e.g., [15,22,23,26,27,32,33]. For applications to algebraic statistics related to binary graph models, Markov random fields and phylogenetic models on split systems as a generalization of binary Jukes-Cantor models, see for example [34].…”