Using $L^*$ to denote the mirror image of a $\mu$-component link $L$, the HOMFLY polynomial satisfies $P_{L^*}(l,m)=P_L(l^{-1},m)$ while the Alexander-Conway polynomial (i.e. a symmetric representative of the Alexander polynomial where the sign is appropriately fixed) satisfies $\Delta_{L^*}(t)=(-1)^{\mu-1}\Delta_L(t)$ (use Seifert matrices).
On the other hand, Cromwell's textbook indicates that $\Delta_L(t)=P_L(1,t^{-1/2}-t^{1/2})$. I understand why this is true (both polynomials satisfy the same skein relation) but I've gotten confused about why this doesn't contradict the above mirror formulas: plugging in $l=1$ in HOMFLY won't allow you to pick up this $(-1)^{\mu-1}$ in Alexander-Conway.
Can anyone help?
The problem is in definitions of HOMFLY skein relations, Your first equation holds when the skein relation is as in equation (3): https://mathworld.wolfram.com/HOMFLYPolynomial.html
The formula to put $l=1$ to obtain the Alexander skein realtion is different (the sign between $l(..)-l^{-1}(..)$ then should be minus).
---- Edit:
I looked up now on Cromwell's textbook, so apparenty there is an error there in Theorem 10.2.3(b)