Let $K$ is a knot and let $\bar{K}$ be the mirror image of $K$.
I want to confirm this relationships.
Let $f_K(t)$ be the Kauffman polynomial of $K$. To get the mirror image we swap every right hand move with a left hand move and vice versa. So every $t \to t^{-1}$ and $t^{-1}\to t$. Thus $f_{\bar{K}}(t)=f_{K}(t^{-1})$.
A similar relationship exists with the HOMFLY polynomial: $P_{\bar{K}}(l,m)=P_{K}(-l,m)$. I am having problems establishing the HOMFLY relationship. Could I get a push in the right direction for that one?