Skein relation for the Jones polynomial - Example not working out

Question

I've decided to learn some knot theory during this summer, using The Knot Book. Today, I showed that the Jones polynomial satisfies the Skein relation $$t^{-1}V(L_+) - tV(L_-) + (t^{-1/2}-t^{1/2})V(L_0)$$.

After showing this, I decided to work out an example. I picked the right-handed trefoil. We have previously calculated that the Jones polynomial of the riht-handed trefoil is $t+t^3-t^4$. All the crossings are negative, so I substituted the Jones polynomial of the trefoil for $V(L_-)$, and the switched the crossing to obtain $L_+$ and $L_0$. I obtained that $L_+$ can be turned into the unknot by a couple of Reidemeister moves, and hence decided that $V(L_+) = 1$. Finally, $L_0$ turns out to be the Hopf link, which has Jones polynomial $-t-t^{-1}$. So I substituted $V(L_0) = -t-t^{-1}$.

Then, looking at my Skein relation, I have:

$$t^{-1}(1)-t(t+t^3-t^4) + (t^{-1/2}-t^{1/2})(-t-t^{-1}) = t^{-1} - t^2 - t^4 + t^5 - t^{1/2} - t^{-3/2} + t^{3/2} + t^{-1/2} \neq 0$$

Why is this not working out correctly? What am I missing? I've checked that my polynomials for the trefoil and the Hopf link is correct, and I've checked my algebra, so I assume I'm missing something more fundamental.

Answer 1

The part you are missing is that mirror images in the Jones polynomial take $t$ and replace it with $t^{-1}$. Also, I suppose you mean the Skein relation at the top equals 0. Another mistake is Hopf Link's Jones Polynomial. If you look at page 15 of Jones' presentation on his polynomial, we see the positive Hopf link has a polynomail of $-t^{\frac{5}{2}}-t^{\frac{1}{2}}$. (Note: It appears that Wolfram's Mathworld is wrong.) But in our problem, we have the negative Hopf Link, which has Jones polynomial $-t^{-\frac{5}{2}}-t^{-\frac{1}{2}}$. And the right handed trefoil, or the mirror image will have Jones polynomial $-t^{-4}+t^{-3}+t^{-1}$. Making these changes we should have:

$$t^{-1}(1)-t\left( -t^{-4}+t^{-3}+t^{-1} \right)+\left(t^{-\frac{1}{2}}-t^{\frac{1}{2}}\right)\left( -t^{-\frac{5}{2}}-t^{-\frac{1}{2}}\right)=0$$

Hope this helps.