  In this paper, we compute special values of certain combinatorial zeta
functions counting geodesic paths in the (n-1)-skeleton of a triangulation of a
n-dimensional manifold. We show that they carry a topological meaning. As such,
we recover the first Betti number and L2-Betti number of compact manifolds, and
the linking number of pairs of null-homologous knots in a 3-manifold. The tool
to relate the two sides (counting geodesics/topological invariants) are random
walks on higher dimensional skeleta of the triangulation.
