  We use Fox calculus to assign a marked polytope to a `nice' group
presentation with two generators and one relator. Relating the marked vertices
to Novikov-Sikorav homology we show that they determine the
Bieri-Neumann-Strebel invariant of the group. Furthermore we show that in many
cases the marked polytope is an invariant of the underlying group and that in
those cases the marked polytope also determines the minimal complexity of all
the associated HNN-splittings.
