2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228079As in the case of the associahedron and cyclohedron, the permutohedron can also be defined as an appropriate compactification of a configuration space of points on an interval or on a circle. The construction of the compactification endows the permutohedron with a projection to the cyclohedron, and the cyclohedron with a projection to the associahedron. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disk, but are still contractible. We briefly explain an application of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss manifold calculus.27 pages The new version gives a more detailed exposition for the projection from the cyclohedron to the associahedron as maps of compactifications of configuration spaces. We also develop a similar picture for the projection from the permutohedron to the cyclohedron/associahedronAlgebraic TopologyGeometric Topology51M20 (Secondary) 57N25, 18D50 (Secondary)text