In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements.
Statement Let O denote the unknot. For any knot K {\displaystyle K} , let ⟨ K ⟩ N {\displaystyle \langle K\rangle _{N}} be the Kashaev invariant of K {\displaystyle K} , which may be defined as
⟨ K ⟩ N = lim q → e 2 π i / N J K , N ( q ) J O , N ( q ) {\displaystyle \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 ) {\displaystyle J_{K,N}(q)} is the N {\displaystyle N} -Colored Jones polynomial of K {\displaystyle K} . The volume conjecture states that
lim N → ∞ 2 π log | ⟨ K ⟩ N | N = vol ( S 3 ∖ K ) {\displaystyle \lim _{N\to \infty }{\frac {2\pi \log |\langle K\rangle _{N}|}{N}}=\operatorname {vol} (S^{3}\backslash K)} , where vol ( S 3 ∖ K ) {\displaystyle \operatorname {vol} (S^{3}\backslash K)} is the simplicial volume of the complement of K {\displaystyle K} in the 3-sphere, defined as follows. By the JSJ decomposition, the complement S 3 ∖ K {\displaystyle S^{3}\backslash K} may be uniquely decomposed into a system of tori
S 3 ∖ K = ( ⨆ i H i ) ⊔ ( ⨆ j E j ) {\displaystyle S^{3}\backslash K=\left(\bigsqcup _{i}H_{i}\right)\sqcup \left(\bigsqcup _{j}E_{j}\right)}
with H i {\displaystyle H_{i}} hyperbolic and E j {\displaystyle E_{j}} Seifert-fibered. The simplicial volume vol ( S 3 ∖ K ) {\displaystyle \operatorname {vol} (S^{3}\backslash K)} is then defined as the sum
vol ( S 3 ∖ K ) = ∑ i vol ( H i ) {\displaystyle \operatorname {vol} (S^{3}\backslash K)=\sum _{i}\operatorname {vol} (H_{i})} , where vol ( H i ) {\displaystyle \operatorname {vol} (H_{i})} is the hyperbolic volume of the hyperbolic manifold H i {\displaystyle H_{i}} . As a special case, if K {\displaystyle K} is a hyperbolic knot, then the JSJ decomposition simply reads S 3 ∖ K = H 1 {\displaystyle S^{3}\backslash K=H_{1}} , and by definition the simplicial volume vol ( S 3 ∖ K ) {\displaystyle \operatorname {vol} (S^{3}\backslash K)} agrees with the hyperbolic volume vol ( H 1 ) {\displaystyle \operatorname {vol} (H_{1})} .
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