The volume conjecture

For a hyperbolic knot \(K\), \(2\pi \lim_{N\to\infty} \log|J_N(K; e^{2\pi i/N})| / N\) equals the hyperbolic volume of the knot complement, where \(J_N\) is the \(N\)-coloured Jones polynomial.

Kashaev (1997) noticed the asymptotics of his quantum-dilogarithm invariant recover hyperbolic volume; Murakami–Murakami (2001) reformulated it via the coloured Jones polynomial, opening the problem to a generation of quantum topologists. The figure-eight knot was verified early; families of knots and links followed (van der Veen, Chen–Yang, Ohtsuki...). A full proof would unite quantum invariants with classical geometry — and explain why a representation-theoretic quantity knows about volume.