Volume conjecture

In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. ⟨ K ⟩ N = lim q → e 2 π i / N J K , N ( q ) J O , N ( q ) . {displaystyle langle K angle _{N}=lim _{q o e^{2pi i/N}}{frac {J_{K,N}(q)}{J_{O,N}(q)}}.}     (1) lim N → ∞ 2 π log ⁡ | ⟨ K ⟩ N | N = vol ⁡ ( K ) , {displaystyle lim _{N o infty }{frac {2pi log |langle K angle _{N}|}{N}}=operatorname {vol} (K),,}     (2) lim N → ∞ 2 π log ⁡ ⟨ K ⟩ N N = vol ⁡ ( S 3 ∖ K ) + C S ( S 3 ∖ K ) , {displaystyle lim _{N o infty }{frac {2pi log langle K angle _{N}}{N}}=operatorname {vol} (S^{3}ackslash K)+CS(S^{3}ackslash K),}     (3) In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. Let O denote the unknot. For any knot K let ⟨ K ⟩ N {displaystyle langle K angle _{N}} be Kashaev's invariant of K {displaystyle K} ; this invariant coincides with the following evaluation of the N {displaystyle N} -colored Jones polynomial J K , N ( q ) {displaystyle J_{K,N}(q)} of K {displaystyle K} :