Where can I read about exponential sums corresponding to Jones Polynomial? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:53:07Z http://mathoverflow.net/feeds/question/99766 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99766/where-can-i-read-about-exponential-sums-corresponding-to-jones-polynomial Where can I read about exponential sums corresponding to Jones Polynomial? unknown (google) 2012-06-16T02:17:52Z 2012-06-17T23:35:45Z <p>I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial: <code>$$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{SU(2) connections on \mathbb S^3}/\text{gauge}}e^{ik\operatorname{CS}(A)}\cdot\operatorname{tr}\operatorname{hol}_KA\cdot\mathcal DA$$</code> 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?</p> <p>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 <em>Gauss linking integral</em> of $K$ with itself (that is, once we give $K$ a framing). I believe the arithmetic counterpart is supposed to be a <em>Gauss sum</em> (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 $\operatorname{Spec}\mathbb Z$.</p> http://mathoverflow.net/questions/99766/where-can-i-read-about-exponential-sums-corresponding-to-jones-polynomial/99863#99863 Answer by Igor Rivin for Where can I read about exponential sums corresponding to Jones Polynomial? Igor Rivin 2012-06-17T23:35:45Z 2012-06-17T23:35:45Z <p>For related stuff, see <a href="http://people.math.gatech.edu/~stavros/publications/degqholonomic.pdf" rel="nofollow">this paper by Garoufalidis.</a> (page 12-ish)</p>