The Jones polynomial at specific values of $t$

The evaluation of the Jones polynomial at $e^{i\pi/3}$ is related to the number of 3-colourings $tri(K)$ of $K$ (see also here) as well as to the topology of the branched double cover $\Sigma(K)$:

$$tri(K) = 3\left|V^2_K(e^{i\pi/3})\right| = 3^{\dim H_1(\Sigma(K);\mathbb{Z}/3\mathbb{Z})+1}$$

This was proved by Przytycki in this paper (Theorem 1.13) and Lickorish-Millet here. I don't know whether similar relations hold for more general Fox colourings.

This is not really an answer to the precise questions you're asking, but it's a pretty result.

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UPDATE (Aug 19, 2014): I have found some more references and some more info in this problem list: the third remark on page 383 (page 11 of the PDF) covers what was known in 2004. In particular, it says that computing $V_K(\omega)$ is $\#P$-hard (see Neil Hoffman's comment below) unless $\omega$ is a power of $e^{i\pi/3}$ or $\omega = \pm i$, and it gives the interpretation for $V_K(\omega)$ in the four remaining cases (the first two have been mentioned by Jim Conant in the comments above). If $L$ is a link, I will call $\ell$ the number of components, and $\Sigma(L)$ the double cover of $S^3$ branched over all components of $L$.

• $V_L(1) = (-2)^{\ell - 1}$; for a knot, $V_K(1) = 1$;
• $\left|V_L(-1)\right| = \left|H_1(\Sigma(L))\right|$ if $H_1(\Sigma(L))$ is torsion, and is 0 otherwise; for a knot, $\left|V_K(-1)\right| = \left|\det(K)\right|$;
• $V_L(i) = (-\sqrt2)^{\ell-1}(-1)^{\mathrm{Arf}(L)}$ if $L$ is a proper link (i.e. ${\rm lk}(K,L\setminus K)$ is even for every component $K$ of $L$), and vanishes otherwise (Murakami); notice that the Arf invariant is defined only for proper links.
• $V_L(e^{2i\pi/3}) = 1$.

The volume conjecture predicts the existence limit of (a certain normalization of) the colored Jones polynomials evaluated at roots of unity (which is not known to exist), and that this limit is equal to the hyperbolic volume of the knot complement. This volume is uniquely defined and in some sense is a "physical" quantity.

(Note: This doesn't literally answer the question, since the answer generalizes "Jones polynomial" to "colored Jones polynomial.")