Jones polynomial
E169187
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Jones polynomial canonical | 21 |
| Jones polynomial of alternating links | 1 |
| colored Jones polynomial | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1483880 — 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: Jones polynomial Context triple: [Conway polynomial, relatedInvariant, Jones polynomial]
-
A.
Conway polynomial
The Conway polynomial is an invariant of knots and links in topology that assigns a polynomial to each knot, capturing essential information about its structure and helping distinguish non-equivalent knots.
-
B.
Conway notation for knots
Conway notation for knots is a mathematical system introduced by John H. Conway that encodes knot and link diagrams into concise symbolic expressions to classify and study them.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- 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: Jones polynomial Target entity description: 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.
-
A.
Conway polynomial
The Conway polynomial is an invariant of knots and links in topology that assigns a polynomial to each knot, capturing essential information about its structure and helping distinguish non-equivalent knots.
-
B.
Conway notation for knots
Conway notation for knots is a mathematical system introduced by John H. Conway that encodes knot and link diagrams into concise symbolic expressions to classify and study them.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Laurent polynomial–valued invariant
ⓘ
knot invariant ⓘ link invariant ⓘ |
| arisesFrom |
representations of braid groups
ⓘ
subfactor theory in von Neumann algebras ⓘ |
| canBeComputedFrom | braid representation of a link ⓘ |
| canBeComputedUsing |
Kauffman bracket
ⓘ
linked to:
Kauffman polynomial
|
| categorifiedBy | Khovanov homology ⓘ |
| codomain |
Laurent polynomials in a variable q^{1/2}
ⓘ
Laurent polynomials in a variable t^{1/2} ⓘ |
| connectedTo |
Chern–Simons topological quantum field theory
ⓘ
linked to:
Chern–Simons theory
Witten–Reshetikhin–Turaev invariants ⓘ
linked to:
Witten–Reshetikhin–Turaev invariant
quantum groups ⓘ |
| definedBy | skein relation at a crossing ⓘ |
| dependsOn | choice of orientation of link components ⓘ |
| distinguishes | many non-equivalent knots ⓘ |
| domain | oriented links in 3-dimensional space ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| generalizationOf | Alexander polynomial in some contexts ⓘ |
| hasCoefficientRing | integers ⓘ |
| hasExponentType | half-integers in the variable exponent ⓘ |
| inspired |
categorification leading to Khovanov homology
ⓘ
development of quantum invariants of 3-manifolds ⓘ |
| introducedIn | paper on representations of braid groups ⓘ |
| invariantUnder |
Reidemeister moves
ⓘ
ambient isotopy of links ⓘ |
| namedAfter | Vaughan Jones ⓘ |
| normalizationCondition | value on the unknot equals 1 ⓘ |
| openProblem | whether the Jones polynomial detects the unknot ⓘ |
| property |
different knots can share the same Jones polynomial
ⓘ
not a complete knot invariant ⓘ |
| refines | classical knot invariants ⓘ |
| relatedConjecture | Volume conjecture ⓘ |
| relatedTo |
HOMFLY polynomial
ⓘ
linked to:
HOMFLY-PT polynomial
HOMFLY-PT polynomial ⓘ Kauffman polynomial ⓘ |
| satisfies |
behavior under connected sum of knots
ⓘ
multiplicativity under disjoint union up to normalization ⓘ skein relation ⓘ |
| usedIn |
distinguishing mirror-image knots in some cases
ⓘ
studying chirality of knots ⓘ topological quantum computation ⓘ |
| usedToDistinguish | trefoil knot from the unknot ⓘ |
| valueOnUnknot | 1 ⓘ |
| variableConvention |
q
ⓘ
t ⓘ |
| yearIntroduced | 1984 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Jones polynomial Description of subject: 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.
Referenced by (23)
Full triples — surface form annotated when it differs from this entity's canonical label.