Where can I read about exponential sums corresponding to Jones Polynomial?

2012-06-16T02:17:52Z

I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial: $$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb S^3$}/\text{gauge}}e^{ik\operatorname{CS}(A)}\cdot\operatorname{tr}\operatorname{hol}_KA\cdot\mathcal DA$$ is some sort of exponential sum. I can't remember the reference, though, and I've found nothing (really absolutely nothing) with Google. Does anyone know of a reference or expository account of this type of analogy?

If it helps (or entertains), I can say what I believe the analogy should be in the "abelian case" when we replace $SU(2)$ with $U(1)$. With $U(1)$ in place of $SU(2)$, the path integral can be "calculated" exactly, and the result is more or less equal to the Gauss linking integral of $K$ with itself (that is, once we give $K$ a framing). I believe the arithmetic counterpart is supposed to be a Gauss sum (is it a coincidence both are associated with Gauss?), which is of course related to the Legrendre symbol $(\frac pq)$, which is in turn interpreted in Mazur's dictionary between three-manifolds and rings of integers in number fields as the linking number of $\operatorname{Spec}\mathbb F_p$ and $\operatorname{Spec}\mathbb F_q$ in $\operatorname{Spec}\mathbb Z$.

Answer by Igor Rivin for Where can I read about exponential sums corresponding to Jones Polynomial?

2012-06-17T23:35:45Z

For related stuff, see this paper by Garoufalidis. (page 12-ish)

Where can I read about exponential sums corresponding to Jones Polynomial? - MathOverflow

Where can I read about exponential sums corresponding to Jones Polynomial?

Answer by Igor Rivin for Where can I read about exponential sums corresponding to Jones Polynomial?