  We calculate the probabilities that a trajectory of a fractional Brownian
motion with arbitrary fractal dimension $d_f$ visits the same spot $n \ge 3$
times, at given moments $t_1, ..., t_n$, and obtain a determinant expression
for these probabilities in terms of a displacement-displacement covariance
matrix. Except for the standard Brownian trajectories with $d_f = 2$, the
resulting many-body contact probabilities cannot be factorized into a product
of single loop contributions. Within a Gaussian network model of a
self-interacting polymer chain, which we suggested recently, the probabilities
we calculate here can be interpreted as probabilities of multi-body contacts in
a fractal polymer conformation with the same fractal dimension $d_f$. This
Gaussian approach, which implies a mapping from fractional Brownian motion
trajectories to polymer conformations, can be used as a semiquantitative model
of polymer chains in topologically-stabilized conformations, e.g., in melts of
unconcatenated rings or in the chromatin fiber, which is the material medium
containing genetic information. The model presented here can be used,
therefore, as a benchmark for interpretation of the data of many-body contacts
in genomes, which we expect to be available soon in, e.g., Hi-C experiments.
