Abstract
The entanglement of polymers in a melt or concentrated solution can be measured by the plateau modulus and takes the form
c
x
where
c
is the effective concentration of polymer and 2 <
x
< 3. Different arguments exist in the literature for
x
. It is shown in this paper that
x
may be directly calculated and that the value 3 comes from purely random walks as noted by Doi in a scaling argument,
x
= 2.25 from semidilute scaling theory, and
x
= 2 from a direct calculation of the entanglement by a cage of rectilinear molecules. Physical reality is reasonably placed between these limits. The whole analysis is based on the Gauss topological invariant.
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