Seminormality, canonical modules, and regularity of cut polytopes

TL;DR

This paper investigates the algebraic properties of cut polytopes, focusing on normality, seminormality, and regularity, providing classifications and bounds for various graph types, and exploring their canonical modules.

Contribution

It offers new insights into the normality and seminormality of cut algebras, computes their regularity for different graphs, and classifies graphs with low regularity.

Findings

Normal cut algebras of simplicial and simple polytopes are Cohen--Macaulay.

The cut algebra of K_5 is not seminormal, implying it is not normal.

Classified graphs with cut algebra regularity ≤ 4.

Abstract

Motivated by a conjecture of Sturmfels and Sullivant we study normal cut polytopes. After a brief survey of known results for normal cut polytopes it is in particular observed that for simplicial and simple cut polytopes their cut algebras are normal and hence Cohen--Macaulay. Moreover, seminormality is considered. It is shown that the cut algebra of $K_5$ is not seminormal which implies again the known fact that it is not normal. For normal Gorenstein cut algebras and other cases of interest we determine their canonical modules. The Castelnuovo-Mumford regularity of a cut algebra is computed for various types of graphs and bounds for it are provided if normality is assumed. As an application we classify all graphs for which the cut algebra has regularity less than or equal to 4.