# Undirected Alexander polynomial (sort of)

Take the skein relation of the Alexander polynomial: $$S^1-S^{-1}-zS^0=0$$, where z is the parameter of the Alexander polynomial and $$S$$ the overcross braid element. "Multiply" the equation with an overcrossing or undercrossing to get new ones. Repeat. Multiply all those equations with a coefficient and add them up. It suffices to get to skein "dimension" 8 to be able to do the following: With the correct choice of coefficients, you can turn around one of the arrows and the equation stays valid. It looks like $$(S^4+S^{-4}-2*S^0)+(S^2+S^{-2}-2*S^0)(z^2+z^{-2})^2=0$$. This is what I call the undirected Alexander polynomial (since it can be computed without directions on a link).
You can now ask "Hey, does that skein equation uniquely define a link polynomial?" but I also have an accompanying S matrix of dimension $$4^4$$ (which is a bit to big to post it here) that fulfills the usual conditions and this equation.
My question is if this polynomial is well known, and especially which Lie algebra is associated with it (like $$A_1$$ with the Kauffman - thus the rt tag).
(Background: I identified all charge-preserving S matrices of dimension $$<=4$$ and the Lie algebras associated, except this one.)