Now my general understands of finding kauffman's bracket polynomial is that smoothing and certain reidemeister moves can be turned into variables (i.e. smoothing a crossing by linking the top and bottom two lines is regarded as an $A$, whereas linking the right and left lines is regarded as $A^{-1}$). However after going through my notes I've noticed these two transformations:
To me these seem like the same transformation, two reidemeister moves 1 (which means uncurling the line), but they are given two different variables, $A{^6}$ and ${A^-6}$. I was just wondering if anyone can explain to me why they are different.
In the first case, you are using the Reidemeister I move to undo two negative twists, and in the second case, you are using the Reidemeister I move to undo two positive twists.
Orient the knot in any way you'd like. Locally (and up to rotation) each crossing will look like one of the two crossings below.
A Reidemeister one move either has a positive crossing or a negative crossing depending on how one twists to form the kink. You can use the usual Kauffman bracket relation
and the relation on disjoint union of crossingless components $$\langle D \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle D \rangle$$ to prove that undoing a positive Reidemeister I twist results in multiplication by $-A^3$ and undoing a negative Reidemeister I twist results in multiplication by $-A^{-3}$.