Yeah, cross product is the determinant with "one strand bent back."
If you take a 3-regular planar graph and put the determinant/Levi-Civita tensor on each vertex, you get plus-or-minus the number of edge-3-colorings of that graph. The four-color theorem is merely to show this tensor contraction is always non-zero :-) (Nobody's managed to prove it this way, yet.)
Here are some notes showing how to contract two Levi-Civita symbols together graphically: https://imgur.com/HmaLSaB.png
This is in Penrose's 1971 paper ("Applications of negative-dimensional tensors"). He went on to notice that you can do a "puff-up-and-contract" construction to completely get rid of all the Levi-Civita tensors: https://imgur.com/eK9ZCkc.png
From here, he recognized that this sort of tensor network on 3-dimensional space was equivalent to an abstract tensor network on "(-2)-dimensional space", where there is an identity that let him remove all the places the strings cross https://imgur.com/65z0zov.png (This element placed at each edge in (-2)-dimensional space is known as the "second Jones-Wenzl projector for the Temperley-Lieb algebra.")
From a more recent point of view, what he did was find a graphical way to go from an SO(3) tensor network to an SL(2) tensor network (which should be possible since both Lie groups are isomorphic).