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I tried computing the Jones polynomial for the left-handed trefoil knot, but ran into a bit of an issue with how I pick my crossings for the L_0 skein relation.

I decided to work with the lower L_+ relation as indicated here:

enter image description here

In the following expression, let P(K) be the Jones polynomial of a knot K in the variable x. Working out the skein relation that the Jones polynomial satisfies produces a result which is incorrect:

enter image description here

(notice that both L_- and L_0 end up working out as the unknot)

However for L_0, there's a different crossing that can be picked, leading to the following knot (the double link):

enter image description here

And thus a new expression:

enter image description here

And a new result, this time the correct one (after I've computed and substituted in the polynomial of the double link).

What I'm struggling with is how to pick a crossing for L_0 here (and in general too). It's not obvious to me at all why I should have gone with the double link instead of the unknot.

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Assign some (any) orientation to the knot, and after you do that, there will be only one choice of $L_0$ that is consistent with the orientation.

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  • $\begingroup$ That was it, thank you, I defined an orientation on the knot and it was quickly obvious that for L_0 the crossing I'd originally picked just wouldn't work. $\endgroup$
    – Nobilis
    Commented Mar 12, 2023 at 18:37

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