A tale of two $q$-deformations : connecting dual polar spaces and weighted hypercubes

Two $q$-analogs of the hypercube graph are introduced and shown to be related through a graph quotient. The roles of the subspace lattice graph, of a twisted primitive elements of $U_q(\mathfrak{s}\mathfrak{u}(2))$ and of the dual $q$-Krawtchouk polynomials are elaborated upon. This paper is dedicated to Tom Koornwinder.