Pensieve Header: In Duzhin’s world, a diagram with a single trivalent vertex need not be 0. This disproves a conjecture in http://katlas.math.toronto.edu/drorbn/bbs/show?shot=LazyKnots-110607-171439.jpg.

F[x_,y_,z_] := Det[(({{x[1], x[2], x[3]}, {y[1], y[2], y[3]}, {z[1], z[2], z[3]}}))];
dot[x_, y_] := x[1]y[1]+x[2]y[2]+x[3]y[3];
Symmetrize[vars_, expr_] := Sum[expr /. Thread[vars -> p], {p, Permutations[vars]}];
AntiSymmetrize[vars_, expr_] := Sum[Signature[p]*expr /. Thread[vars -> p], {p, Permutations[vars]}];

Factor[
Symmetrize[{a,b,c,d,e}, F[a,b,c]dot[b,d]dot[c,e]]
]

Factor[
Symmetrize[{a,b,c,d,e,f}, F[a,b,c]dot[b,d]dot[c,e]dot[e,f]]
]