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Polymer rings in solution are either permanently entangled or not. Permanenttopological restrictions give rise to additional entropic interactions apartfrom the ones arising due to mere chain flexibility or excluded volume.Conversely, entangled polymer rings systems may be formed by closing randomlyentangled flexible linear chains. The dependance of linking numbers betweenrandomly entangled rings on the chain length, more specifically the secondtopological moment $<n^2>$, i.e. the average squared linking number, may bedetermined. In this paper, an approach recently discussed in mathematicalphysics and called abelian BF theory, is presented which allows to express thelinking constraint in its simplest form, the Gauss integral, in terms of twogauge fields. The model of Brereton and Shah for a single ring entangled withmany other surrounding rings is rederived. The latter model is finally used tocalculate the second topological moment, in agreement with a recent result byFerrari, Kleinert, and Lazzizzera obtained by using ${\rm n}$-component$\phi^4$ theory in the limit ${\rm n}\to 0$.Comment: 11 pages, Latex, accepted for publication in J.Phys.

The direct link model for polymer rings using topological field theory and the second topological moment in dense systems