The Jones polynomial at specific values of $t$

Question

I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.

For example, if I take the Jones polynomial for the (right) Trefoil knot, I have

$J(t) = t + t^3 - t^4$.

Is there some way I can interpret $J(0)$? $J(1)$?

I understand that the Jones polynomial is a laurent polynomial, so I don't expect $J(0)$ to make sense for a lot of knots (for example the left trefoil has $J(t) = t^{-1} + t^{-3} - t^{-4}$), but I thought it was worth asking.

I also know that $J(t^{-1})$ gives the Jones polynomial of the mirror image knot. Is there a way to interpret $J(-t)$? $J(t^2)$? How about $J(t) = 0$?

Edit to clarify what I mean when I say "physical meaning": Since the Jones polynomial is a link invariant, $J(0)$ is also a link invariant (if it exists). Does this invariant correspond to a property of the knot that you can visualise, such as, say, the linking number or the crossing number?

Answer 1

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.
• $V_L(e^{2i\pi/3}) = 1$.

Answer 2

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.")