  The field theory approach to the statistical mechanics of a system of N
polymer rings linked together is generalized to the case of links that have a
fixed number $2s$ of maxima and minima. Such kind of links are called plats and
appear for instance in the DNA of living organisms. The topological states of
the link are distinguished using the Gauss linking number. This is a relatively
weak link invariant in the case of a general link, but its efficiency improves
when $2s-$plats are considered. It is proved that, if we restrict ourselves to
$2s-$plat conformations, the field theoretical model established here is able
to take into account also the interactions of topological origin involving
three chains simultaneously. It is shown that these three-body interactions
have nonvanishing contributions when three or more rings are entangled
together, enhancing for instance the attractive forces between monomers. The
model can be used to study the statistical mechanics of polymers in confined
geometries, for instance when $2s$ extrema of a few polymer rings are attached
to membranes. Its partition function is mapped here into that of a multi-layer
electron gas. Such quasi-particle systems are studied in connection with
several interesting applications, including high-$T_c$ superconductivity and
topological quantum computing. At the end an useful connection with the
cosh-Gordon equation is shown.
