The Algebraic and Topological Structure of the Space of Minimal Trajectories in the Hypercube Q4

We study the algebraic and geometric structure of the space of minimal trajectories in the discrete hypercube Q₄. We show that the set of minimal trajectories between opposite vertices can be naturally identified with the symmetric group S₄, where each trajectory corresponds to a permutation describing the order of elementary coordinate changes. Endowed with its natural adjacency relation, this space coincides with the Cayley graph of S₄ with respect to adjacent transpositions, and is isomorphic to the edge graph of the permutahedron P₄. This establishes a structural correspondence between minimal trajectories, permutations, and vertices of this polytope, providing a unified framework connecting discrete geometry, algebraic combinatorics, and group theory.