Homology of the circle with non-trivial local coefficients is trivial. From
this well-known fact we deduce geometric corollaries concerning links of
codimension two. In particular, the Murasugi-Tristram signatures are extended
to invariants of links formed of arbitrary oriented closed codimension two
submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds
are not assumed to be disjoint, but are transversal to each other, and the
signatures are parametrized by points of the whole torus. Murasugi-Tristram
inequalities and their generalizations are also extended to this setup.