  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.
