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 closing 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, called the minimally-interfering closure, can be profitably used to characterize the arc entanglement in a robust and computationally-efficient way. This closure method is applied to the knot localization problem which is tackled using two different localization schemes based on top-down or bottom-up searches.
