If we let $F_{L}(t)$ denote the Kauffman polynomial and $P_{L}(x,y)$ denote the HOMFLY polynomial, then we can obtain the Kauffman polynomial from the HOMFLY polynomial using the following substitution: $$ x= it^{4} \,\text{and } y=i(t^2-t^{-2}).$$
Looking at the Skein relations we have: $xP_{L_{+}}(x,y)+x^{-1}P_{L_{-}}(x,y)+yP_{L_s}(x,y)=0$
The substituting stuff in: $it^4P_{L_{+}}(it^4,i(t^2-t^{-2}))-it^4P_{L_{-}}(it^4,i(t^2-t^{-2}))+i(t^2-t^{-2})P_{L_s}(it^4,i(t^2-t^{-2}))=0$
Does it just follow from the fact that now all the polynomials have become a single variable $t$, so we can just rename them as $P_{L_{+}}(it^4,i(t^2-t^{-2}))$ to $F_{L_+}(t)$ and so forth?
Is is really this easy, or is there something deeper with these substitutions?