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THE PROBABILITY THEORY OF 3-MANIFOLDS. (2 of 4)
Fix the number n of tetrahedra. Then there are finitely many
triangulations of M having n
tetrahedra. Associating
to every such triangulation the maximal mean commute time of the
random walk on it defines
a random variable on the finite, uniform probability space
of triangulations of M
with n tetrahedra.
Our main results are empirical
density functions of these random variables, for certain
manifolds and certain
interesting values of n. The densities are calculated by executing
computer software developed by
the author. Our algorithm generates a large number of
triangulations of a fixed
manifold at random and then calculates mean commute times for
random walks on each of these
triangulations by computing the effective electrical
resistance of an associated
quartic graph, using classical results from graph theory.
As a by-product of these
investigations we define a topological invariant, the electrical
resistance of a 3-manifold,
which we interpret as a refined complexity measure of the
manifold. For example, the
3-sphere has resistance 0.4 and the 3-dimensional Klein bottle
has resistance 0.8056.
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