At a crossing of a knot, if I change the crossing by swapping the two lines, the knot is changed, along with its Jones Polynomial. Witten's path integral

$$ \int {D \mathcal{A}\ e^{i\mathcal{L}}\ W_R(C)}, $$

computes the Jones polynomial and hence should be different when something like this happens. Is there a simple argument / a way to see that this path integral will change under such an operation?

I feel this question is really about the intuition of why Witten's approach works.

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    $\begingroup$ you may want to be a bit more specific, people who know what you are speaking about maybe will understand your question. however as a person who does not know about the subject i would not even know what to look for to find more about it $\endgroup$
    – lucabtz
    Feb 25 at 12:56
  • $\begingroup$ @lucabtz I made some effort to make the question more clear. However, I probably don't understand the subject well enough to formulate my question clear enough. $\endgroup$
    – Alex
    Feb 25 at 13:19

1 Answer 1


I don't think there is any simpler way than to make the conection with conformal blocks and the Knizhnik-Zamolodchikov equation. If you are comming from the maths side, and know enough to understand the Jones polynomial, I'd recommend "Conformal Field Theory and Topology" by Toshitake Kohno. It's quite thin and very much to the point. Even as a physicist I found it easier going than di Francesco et al.


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