Let
O denote the
unknot. For any knot K, let \langle K \rangle_N be the
Kashaev invariant of K, which may be defined as :\langle K \rangle_N=\lim_{q\to e^{2\pi i/N}}\frac{J_{K,N}(q)}{J_{O,N}(q)}, where J_{K,N}(q) is the N-
Colored Jones polynomial of K. The volume conjecture states that :\lim_{N\to\infty} \frac{2\pi\log |\langle K \rangle_N|}{N} = \operatorname{vol}(S^3 \backslash K), where \operatorname{vol}(S^3 \backslash K) is the
simplicial volume of the complement of K in the
3-sphere, defined as follows. By the
JSJ decomposition, the complement S^3 \backslash K may be uniquely decomposed into a system of tori :S^3 \backslash K = \left( \bigsqcup_i H_i \right) \sqcup \left( \bigsqcup_j E_j \right) with H_i
hyperbolic and E_j
Seifert-fibered. The
simplicial volume \operatorname{vol}(S^3 \backslash K) is then defined as the sum :\operatorname{vol}(S^3 \backslash K) = \sum_i \operatorname{vol}(H_i), where \operatorname{vol}(H_i) is the
hyperbolic volume of the hyperbolic manifold H_i. As a special case, if K is a
hyperbolic knot, then the JSJ decomposition simply reads S^3 \backslash K = H_1, and by definition the simplicial volume \operatorname{vol}(S^3 \backslash K) agrees with the hyperbolic volume \operatorname{vol}(H_1). == History ==