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I am trying to show that if we define a new invariant of knots $W(L)$ which follows the same rules as the Jones polynomial $V(L)$, so that it has value 1 on the unknot and satisfies the Skein relation $$t^{-1}W(L_+) - tW(L_-) + (t^{-1/2}-t^{1/2})W(L_0)=0$$ then we must have $W(L)= V(L)$.

I know some things which I think may come in handy; if we set $t=1$ in the Skein relation, we get $W(L_+)=W(L_-)$. We can change any link diagram to a diagram of trivial links by changing crossings from over to under and vice versa. And the Jones polynomial of a union of $k$ trivial links is $(t^{-1/2}-t^{1/2})^{k-1}$. I feel I should be able to put these together to get the result, but I'm not sure how.

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  • $\begingroup$ Use the skein relations and the Reidemeister moves to reduce the computation to that of the unknot. $\endgroup$
    – anomaly
    Mar 26, 2019 at 13:18
  • $\begingroup$ @anomaly how would I begin to do this? $\endgroup$
    – user657992
    Mar 26, 2019 at 13:36
  • $\begingroup$ Exactly like that. Applying the skein relation to a given knot gives an expression for $V(L)$ in terms of a knot with one fewer crossing and one in which the crossing is reversed; the Reidemeister moves simply the latter. $\endgroup$
    – anomaly
    Mar 26, 2019 at 13:44
  • $\begingroup$ @user848484 What is your version of the Jones polynomial? The Kauffman bracket with writhe normalization? I might try to denormalize $W$ (so that it depends on writhe in exactly the same way as the Kauffman bracket) and then show that, given a $W$ that satisfies this skein relation (if one existed) the denormalized version calculates the Kauffman bracket. The converse is that you can show the Jones polynomial satisfies this skein relation. (Chapter 3 of Lickorish's book shows at least the converse.) $\endgroup$
    – Kyle Miller
    Mar 26, 2019 at 14:19
  • $\begingroup$ @KyleMiller I know of the Kauffman bracket definition of the Jones polynomial, but I hadn't thought of using it for this question, I've been trying to do so just from the rules given for the Jones poly and this new invariant. Is this possible? $\endgroup$
    – user657992
    Mar 26, 2019 at 14:22

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