Kashaev–Murakami–Murakami volume conjecture for knots and links

Let $K$ be a framed knot or link in $S^3$. Let $V_N(K)$ be the colored Jones polynomial corresponding to the $(N+1)$-dimensional irreducible representation of the quantum group $\mathcal{U}_q(sl_2)$, normalized as in the source, and set

$$J_{N-1}(K)=\frac{V_{N-1}(K)}{V_{N-1}(\bigcirc)},\qquad q=\exp(\pi i/N).$$

Here $||S^3\setminus K||$ denotes Gromov's simplicial volume of the complement, and $v_3$ is the hyperbolic volume of the regular ideal tetrahedron. Volume conjecture. For a knot or link $K$,

$$2\,\pi\lim_{N\to\infty}\frac{\log\left|J_{N-1}(K)\right|}{N}=v_3\,||S^3\setminus K||.$$

For hyperbolic knots and links, the right-hand side equals the hyperbolic volume of the complement. The conjecture relates the asymptotic growth of the colored Jones polynomial to geometric volume and is presented here as a conjectural statement; its resolution status is not specified in the supplied text.