The volume conjecture, in knot theory, is an open problem relating the quantum invariants of a knot to the hyperbolic geometry of its complement.
Facts
StatementFor a hyperbolic knot K, the exponential growth rate of the Kashaev invariant of K, taken in the limit as the invariant's index N goes to infinity, equals 2 pi times the simplicial volume of the knot complement S cubed minus K. 1 Proposed Year Progress Toward ResolutionThe conjecture has been verified for specific cases including the figure eight knot (Ekholm), the trefoil and three-twist knot (Kashaev and Yokota), the Borromean rings (Garoufalidis and Le), torus knots (Kashaev and Tirkkonen), twisted Whitehead links (Zheng), Whitehead doubles of nontrivial torus knots, and all knots and links with volume zero (van der Veen). It remains open for general knots and is known to be false for arbitrary links. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Volume Conjecture (Wikipedia)
Statement section
lim N → ∞ 2π log |⟨K⟩_N|/N = vol(S³∖K)
History section
Kashaev stated the formula of the volume conjecture in the case of hyperbolic knots in 1997.
Current status section
The volume conjecture is open for general knots, and it is known to be false for arbitrary links.
- Lead section
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