Conway polynomial
E29419
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Conway polynomial canonical | 5 |
| Alexander–Conway polynomial | 1 |
| Conway normalized Alexander polynomial | 1 |
| Conway–Alexander polynomial | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T231137 — 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: Conway polynomial Context triple: [John H. Conway, notableWork, Conway polynomial]
-
A.
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.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
D.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
E.
Polynomial Root Finder
Polynomial Root Finder is a TI-84 Plus calculator application that computes the roots of polynomial equations quickly and accurately.
- 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: Conway polynomial Target entity description: 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.
-
A.
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.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
D.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
E.
Polynomial Root Finder
Polynomial Root Finder is a TI-84 Plus calculator application that computes the roots of polynomial equations quickly and accurately.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
knot invariant
ⓘ
link invariant ⓘ polynomial invariant ⓘ |
| alsoKnownAs |
Conway polynomial
ⓘ
surface form:
Conway normalized Alexander polynomial
Conway polynomial ⓘ
surface form:
Conway–Alexander polynomial
|
| appliesTo | oriented links with any number of components ⓘ |
| canBeComputedBy |
Seifert matrix methods
ⓘ
skein relation ⓘ |
| captures |
information about knot chirality in some cases
ⓘ
information about knot linking for links ⓘ |
| codomain | Laurent polynomials in one variable ⓘ |
| coefficientInterpretation | lowest-degree nonzero coefficient relates to linking numbers for links ⓘ |
| coefficientProperty | coefficients are integers ⓘ |
| computationalComplexity | can be computed in polynomial time for fixed crossing number but is generally hard for large diagrams ⓘ |
| definedBy | Conway skein triple (L₊, L₋, L₀) ⓘ |
| definedOn | isotopy classes of oriented links in S³ ⓘ |
| degreeProperty | degree of ∇(K) is bounded above by twice the genus of K ⓘ |
| dependsOn | oriented link diagram ⓘ |
| doesNotCompletelyClassify | knots ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| firstCoefficientProperty | constant term is 1 for knots ⓘ |
| firstNontrivialCoefficient | often encodes Arf invariant mod 2 for knots ⓘ |
| functorialityProperty | invariant under ambient isotopy ⓘ |
| generalizes | Alexander polynomial normalization ⓘ |
| inspired | later skein-theoretic definitions of other knot polynomials ⓘ |
| introducedIn | 1960s ⓘ |
| invariantOf |
oriented knots
ⓘ
oriented links ⓘ |
| isNot | complete knot invariant ⓘ |
| namedAfter |
John H. Conway
ⓘ
surface form:
John Horton Conway
|
| normalizationChoice | gives Alexander polynomial with symmetric normalization ⓘ |
| normalizationCondition | ∇(unknot) = 1 ⓘ |
| orientationProperty | independent of choice of orientation up to sign changes in variable ⓘ |
| relatedInvariant |
HOMFLY-PT polynomial
ⓘ
Jones polynomial ⓘ |
| relatedTo | Alexander polynomial ⓘ |
| relationToAlexanderPolynomial | Δ_K(t) = ∇_K(t^{1/2} − t^{−1/2}) up to normalization ⓘ |
| satisfies | skein relation ∇(L₊) − ∇(L₋) = z ∇(L₀) ⓘ |
| symmetryProperty | ∇(K)(z) = ∇(K)(−z) for many knots (evenness property related to Alexander polynomial) ⓘ |
| usedFor |
distinguishing non-equivalent knots
ⓘ
studying knot concordance ⓘ studying link splitting properties ⓘ |
| valueOnSplitUnion | for split union L₁ ⊔ L₂, ∇(L₁ ⊔ L₂) = 0 ⓘ |
| valueOnTrivialLink | for n-component trivial link with n>1, ∇ = 0 ⓘ |
| variable | z ⓘ |
| zeroCondition | vanishes for split links with more than one component ⓘ |
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: Conway polynomial Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
John Horton Conway
this entity surface form:
Conway normalized Alexander polynomial
this entity surface form:
Conway–Alexander polynomial
subject surface form:
John H. Conway
subject surface form:
John H. Conway
this entity surface form:
Alexander–Conway polynomial