As 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.

We investigate correlation between gauge coupling unification, fermion mass spectrum, proton decay, perturbativity and ultraviolet cutoff within an SU(5) grand unified theory with minimal scalar content and an extra adjoint representation of fermions. We find strong correlation between the upper bound on the mass of both the bosonic and fermionic SU(2) triplets and the cutoff. The upper bound on the mass of fermionic triplet responsible for Type III seesaw mechanism is 102.1 GeV for the Planck scale cutoff. In that case both the idea of grand unification and nature of seesaw mechanism could be tested at future collider experiments through the production of those particles. Moreover, the prediction for the proton decay lifetime is at most an order of magnitude away from the present experimental limits. If the cutoff is lowered these predictions change significantly. In the most general scenario, if one does (not) neglect a freedom in the quark and lepton mixing angles, the upper bound on the fermionic triplet mass is at 105.4 GeV (1010 GeV). Since the predictions of the model critically depend on the presence of the higher-dimensional operators and corresponding cutoff we address the issue of their possible origin and also propose alternative scenarios that implement the hybrid seesaw framework of the original proposal.