Volume conjecture

E656686

conjecture in knot theory conjecture in quantum topology mathematical conjecture

The Volume conjecture is a proposed deep link between quantum knot invariants and hyperbolic geometry, asserting that the asymptotic behavior of the colored Jones polynomial of a knot determines the hyperbolic volume of its complement.

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Statements (50)

Predicate	Object
instanceOf	conjecture in knot theory ⓘ conjecture in quantum topology ⓘ mathematical conjecture ⓘ
appliesTo	hyperbolic knots in S^3 ⓘ links with hyperbolic complements ⓘ
asserts	asymptotic growth rate of colored Jones polynomial determines hyperbolic volume of knot complement ⓘ
basedOn	Kashaev invariant conjecture NERFINISHED ⓘ
field	hyperbolic geometry ⓘ knot theory ⓘ low-dimensional topology ⓘ quantum invariants of knots and 3-manifolds ⓘ quantum topology ⓘ
generalFormulationInvolves	colored Jones polynomial evaluated at roots of unity ⓘ
hasVariant	complex volume conjecture NERFINISHED ⓘ generalized volume conjecture NERFINISHED ⓘ volume conjecture for 3-manifolds ⓘ volume conjecture for links ⓘ
implies	deep connection between quantum invariants and hyperbolic geometry ⓘ
influenced	development of quantum topology ⓘ study of asymptotics of quantum invariants ⓘ
isSpecialCaseOf	conjectural relations between Chern–Simons theory and hyperbolic geometry ⓘ
originalFormulationInvolves	Kashaev invariant of a knot NERFINISHED ⓘ hyperbolic volume of knot complement ⓘ limit of N-th Kashaev invariant as N tends to infinity ⓘ
relatesConcept	A-polynomial of a knot ⓘ AJ conjecture NERFINISHED ⓘ Chern–Simons theory NERFINISHED ⓘ Gromov norm NERFINISHED ⓘ Jones polynomial NERFINISHED ⓘ Kashaev invariant NERFINISHED ⓘ asymptotic expansion ⓘ character variety of a knot group ⓘ colored Jones invariants ⓘ colored Jones polynomial NERFINISHED ⓘ hyperbolic 3-manifold ⓘ hyperbolic structure on knot complement ⓘ hyperbolic volume ⓘ knot complement ⓘ quantum dilogarithm ⓘ quantum knot invariants ⓘ quantum parameter q ⓘ quantum topology–geometry correspondence ⓘ root of unity ⓘ simplicial volume ⓘ slope conjecture NERFINISHED ⓘ
statedBy	Hitoshi Murakami NERFINISHED ⓘ Jun Murakami NERFINISHED ⓘ Rinat Kashaev NERFINISHED ⓘ
status	open problem ⓘ
yearProposed	1997 ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jones polynomial → relatedConjecture → Volume conjecture ⓘ