the crossing on one boundary disc of the pants (see last
lesson) the trefoil turns into two distinct trivalent ribbon vertices. Recall
picture of Khovanov homology for the trefoil knot associates a parity cube
to all possible smoothings of the three crossings. So once again we start with the Stasheff associahedron and obtain the
cube. Moreover, this cube gives us an invariant for the trefoil knot
For knots with more crossings, Khovanov homology requires higher dimensions, but maybe we can squeeze those into three dimensions by looking at more complicated polytopes with more square faces.
For example, recall
the 6 crossing
knot which we drew on the Klein quartic surface. Can we obtain this knot from the permutohedron, which has six square faces?