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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 $\operatorname{Spec}\mathbb Z$.

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    $\begingroup$ Are you looking for an invariant associated to a prime in a number field, or for an expression for the Jones polynomial of a knot in $S^3$? For the latter, the number-theoretic limits giving the Jones polynomial are interesting; see front.math.ucdavis.edu/0211.5044 . For the former, AFAIK no one has a convincing analogue of the Jones polynomial. Milnor invariants can be interpreted and the Alexander polynomial is supposed to be related to Iwasawa theory (arxiv.org/abs/0904.3399v1), but I believe there's no good analogue of finite-type invariants with more loops. $\endgroup$
    – Dylan Thurston
    Commented Jun 16, 2012 at 4:52
  • $\begingroup$ I'm looking for something number theoretic (motivated by topology/physics, but not directly related to it). I do remember there is no convincing analogue of the Jones polynomial for primes in a number field. I thought I read somewhere though about exponential sums being a rough analogue of functional integration, and that one got "interesting" exponential sums which roughly correspond to the path integral giving the Jones polynomial. $\endgroup$
    – John Pardon
    Commented Jun 16, 2012 at 19:29
  • $\begingroup$ @DylanThurston the link in your comment is broken, here's a replacement: arxiv.org/abs/math/0211044 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 1:53

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

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