  The interplay between the topological and geometrical properties of a polymer
ring can be clarified by establishing the entanglement trapped in any portion
(arc) of the ring. The task requires to close the open arcs into a ring, and
the resulting topological state may depend on the specific closure scheme that
is followed. To understand the impact of this ambiguity in contexts of
practical interest, such as knot localization in a ring with non trivial
topology, we apply various closure schemes to model ring polymers. The rings
have the same length and topological state (a trefoil knot) but have different
degree of compactness. The comparison suggests that a novel method, termed the
minimally-interfering closure, can be profitably used to characterize the arc
entanglement in a robust and computationally-efficient way. This closure method
is finally applied to the knot localization problem which is tackled using two
different localization schemes based on top-down or bottom-up searches.
