Alexander polynomial
E665082
The Alexander polynomial is a classical knot invariant in algebraic topology that assigns a Laurent polynomial to each knot or link, capturing essential information about its topological structure.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Alexander polynomial canonical | 1 |
| multivariable Alexander polynomial | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7450466 — 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: Alexander polynomial Context triple: [HOMFLY-PT polynomial, generalizes, Alexander polynomial]
-
A.
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.
-
B.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
C.
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.
-
D.
HOMFLY-PT polynomial
The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
-
E.
Arf invariant
The Arf invariant is an algebraic invariant in topology and quadratic form theory that classifies certain quadratic forms over fields of characteristic two and plays a key role in knot theory and surgery theory.
- 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: Alexander polynomial Target entity description: The Alexander polynomial is a classical knot invariant in algebraic topology that assigns a Laurent polynomial to each knot or link, capturing essential information about its topological structure.
-
A.
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.
-
B.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
C.
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.
-
D.
HOMFLY-PT polynomial
The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
-
E.
Arf invariant
The Arf invariant is an algebraic invariant in topology and quadratic form theory that classifies certain quadratic forms over fields of characteristic two and plays a key role in knot theory and surgery theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Laurent polynomial-valued invariant
ⓘ
knot invariant ⓘ link invariant ⓘ |
| appliesTo | links with multiple components ⓘ |
| canDistinguish | some non-equivalent knots ⓘ |
| cannotDistinguish | all non-equivalent knots ⓘ |
| captures |
topological information about a knot
ⓘ
topological information about a link ⓘ |
| category | classical knot invariant ⓘ |
| codomain | Z[t,t^{-1}] ⓘ |
| computableFrom |
Seifert surface of the knot
ⓘ
Wirtinger presentation of the knot group ⓘ |
| definedUsing |
Alexander module
ⓘ
Fox calculus ⓘ Seifert matrix ⓘ determinant of V - tV^T ⓘ first homology of the infinite cyclic cover ⓘ infinite cyclic cover of the knot complement ⓘ presentation matrix of the Alexander module ⓘ |
| degreeRelatedTo | twice the genus for fibered knots ⓘ |
| dependsOn | choice of knot or link ⓘ |
| evaluationProperty | Δ_K(1) = ±1 for a knot ⓘ |
| extension | multivariable Alexander polynomial for links ⓘ |
| field |
algebraic topology
ⓘ
knot theory ⓘ |
| generalizedBy |
higher-order Alexander invariants
ⓘ
twisted Alexander polynomial ⓘ |
| hasAlexanderPolynomial | 1 ⓘ |
| historicalPeriod | introduced in the 1920s ⓘ |
| input |
oriented knot
ⓘ
oriented link ⓘ |
| introducedBy | James Waddell Alexander II ⓘ |
| invariantUnder |
Reidemeister moves
ⓘ
ambient isotopy of knots ⓘ |
| namedAfter | James Waddell Alexander II ⓘ |
| normalizationProperty | defined up to multiplication by ±t^n ⓘ |
| output | Laurent polynomial in one variable ⓘ |
| relatedInvariant |
Conway polynomial
ⓘ
HOMFLY-PT polynomial ⓘ Jones polynomial ⓘ |
| specialCaseOf |
multivariable Alexander polynomial
ⓘ
linked to:
Alexander polynomial
|
| symmetryProperty | Δ_K(t) = ± t^n Δ_K(t^{-1}) ⓘ |
| usedToStudy |
3-manifold topology
ⓘ
fibered knots ⓘ knot concordance ⓘ knot genus ⓘ |
| valueFor | unknot ⓘ |
| variable | t ⓘ |
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: Alexander polynomial Description of subject: The Alexander polynomial is a classical knot invariant in algebraic topology that assigns a Laurent polynomial to each knot or link, capturing essential information about its topological structure.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Alexander polynomial