Paper detail

"Nonconstant cohomology" of Hietarinta's two-color solutions to four-simplex equation

"Nonconstant cohomologies" are introduced for solutions of set-theoretical four-simplex equation (FSE). While usual cohomologies lead to solutions of constant quantum FSE, our "nonconstant cohomologies" lead to solutions of nonconstant quantum FSE. Computer calculations are presented showing that large spaces of such cohomologies exist for all Hietarinta's two-color linear solutions to set-theoretical FSE. After taking a partial trace of the corresponding quantum operators, combined with one additional trick, this leads to solutions of tetrahedron equation, including those with non-negative matrix elements, and not reducible to a permutation, even with cocycle multipliers.