HOMFLY polynomial of hopf link

Question

I have this Hopf link https://i.stack.imgur.com/8mCxI.jpg and when I try to calculate the HOMFLY polynomial I get the standard answer but with negative powers of l i.e. ml^{-1} + m^{-1}(l^{-1}+l^{-3}). Is this because of the specific orientation or have I made a mistake?

Thanks

Answer 1

Using $\ell P(L_+)+\ell^{-1} P(L_-)+m P(L_0)=0$ and $P(\text{unknot})=1$, I get $\ell^{-3}m^{-1}+\ell^{-1}m^{-1}-\ell^{-1}m$ for the Hopf link.

This uses that the HOMFLY polynomial for the split link of two unknots is $\frac{-\ell-\ell^{-1}}{m}$, which either comes from the split link formula or a quick calculation:

HOMFLY of the Hopf link with one of the components reversed in orientation gives the same polynomial but with $\ell$ and $\ell^{-1}$ swapped.