The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ Here we are assuming that the complement of $K$ is hyperbolic, and $vol(K)$ is its hyperbolic volume. (The normalization is that $J_N(q) = 1$ for the unknot.) I have an ignorant question about this:

Q: Is it obvious that this limit exists?

One thing that might be helpful for showing the limit exists is that the $J_n(q)$ satisfy a (generalized) recursion relation (this is a theorem of Garoufalidis and Le). More precisely, there are Laurent polynomials $a_i(-,-)$ such that $$ \sum_i a_i(q,q^N) J_{N+i}(q) = 0$$ (This is true for any $N$, and the $a_i$ don't depend on $N$.) But the existence of this recursion relation was proved several years after the volume conjecture.


No, this is unknown. There are heuristic arguments for convergence based on the stationary phase approximation, but as far as I know, no one has made the argument precise in general. The closest I know of to a proof of convergence is an upper bound on the limsup in terms of the crossing number given in Theorem 1.3 of this paper by Garoufalidis and Le, although I should add the caveat that I haven't been following the developments on this topic closely recently.


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