  Principal circle bundle over a PL polyhedron can be triangulated and thus
obtains combinatorics. The triangulation is assembled from triangulated circle
bundles over simplices. To every triangulated circle bundle over a simplex we
associate a necklace (in combinatorial sense). We express rational local
formulas for all powers of first Chern class in the terms of mathematical
expectations of parities of the associated necklaces. This rational parity is a
combinatorial isomorphism invariant of triangulated circle bundle over simplex,
measuring mixing by triangulation of the circular graphs over vertices of the
simplex. The goal of this note is to sketch the logic of deduction these
formulas from Kontsevitch's cyclic invariant connection form on metric
polygons.
