TITLE: The Jones polynomial at specific values of $t$
QUESTION [32 upvotes]: 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?

REPLY [7 votes]: 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.")