I'm trying to use the relationship between the Skein relations and the Alexander polynomial of a knot:
$$ \Delta_{L+}(t) = \Delta_{L-}(t) + (t^{1/2} - t^{-1/2})\Delta_{L_0}(t) $$
to find the Alexander polynomial of the Figure 8 knot, denoted $K$. In the region I've identified with a red circle in the attached picture, $K=L+$, the unknot is $L-$, and $L_0$ is a trefoil knot. The conventions I've used for $L+$ are ``bottom-left to top-right strand goes on top", $L-$ is `bottom-left to top-right strand goes on bottom,' and $L0$ is the two strands are separated. These conventions should give the Alexander polynomial for the figure 8 knot, which I believe is $-t^{-1}+3-t$, given that the Alexander polynomial for the trefoil knot is $t^{-1}-1+t$. Can you please spot where I've missteped: the polynomial I get contains $t^{3/2}$, for instance, and shouldn't.
Adding in orientations allows one to keep track of Skein relations, and have them work out in the right way with the Alexander polynomial.
Thanks