Volume conjecture
E656686
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Volume conjecture canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338472 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Volume conjecture Context triple: [Jones polynomial, relatedConjecture, Volume conjecture]
-
A.
Witten–Reshetikhin–Turaev invariant
The Witten–Reshetikhin–Turaev invariant is a quantum invariant of 3-manifolds and links derived from Chern–Simons theory and quantum groups, playing a central role in low-dimensional topology and quantum topology.
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B.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
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C.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
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D.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
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E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Volume conjecture Target entity description: 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.
-
A.
Witten–Reshetikhin–Turaev invariant
The Witten–Reshetikhin–Turaev invariant is a quantum invariant of 3-manifolds and links derived from Chern–Simons theory and quantum groups, playing a central role in low-dimensional topology and quantum topology.
-
B.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
C.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
-
D.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
-
E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Volume conjecture Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.