The volume conjecture for knots

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The volume conjecture for knots

   Let $K$ be a knot. The $N$-colored Jones invariant of $K$ is denoted by $J_N(K)$, and let $\mathrm{Vol}(\mathbb{S}^3-K)$ denote the simplicial volume of its complement.

   Volume conjecture. For any knot $K$,

   $$\lim_{N\to \infty}\frac{2\pi}{N}\log\left|J_N(K)\left(e^{\frac{\pi i}{2N}}\right)\right|=\mathrm{Vol}(\mathbb{S}^3-K).$$

   This conjecture proposes a relation between colored Jones invariants at roots of unity and the simplicial volume of a knot complement; its status is not resolved in the supplied source.

   source: Roland van der Veen, “The volume conjecture for augmented knotted trivalent graphs”, arXiv:0805.0094 (2009).

2. The volume conjecture for knots

   Let $K$) be a knot. Write $J_N(K)$ for its normalized $N$-colored Jones polynomial, and let $

   \mathrm{Vol}(K)=v_3\,\|\mathbb{S}^3-K\|$

   where $v_3$ is the volume of the regular ideal hyperbolic tetrahedron and $\|\mathbb{S}^3-K\|$ is the simplicial volume of the complement.

   Volume conjecture. For every knot $K$,

   $$\lim_{N\to\infty}\frac{2\pi}{N}\log\left|J_N(K)\left(e^{\frac{2\pi i}{N}}\right)\right|=\mathrm{Vol}(K).$$

   The paper proves this conjecture for the Whitehead link and related Whitehead chains in the cases described in its main theorem, while also exhibiting non-splittable links for which the original formulation fails.

   source: Roland van der Veen, “Proof of the volume conjecture for Whitehead chains”, arXiv:math/0611181 (2007).