Arf invariant
E654172
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Arf invariant canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T7281888 — 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: Arf invariant Context triple: [Cahit Arf, knownFor, Arf invariant]
-
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.
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B.
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.
-
C.
Milnor number
The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
-
D.
Donaldson invariants
Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
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E.
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.
- 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: Arf invariant Target entity description: 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.
-
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.
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.
-
C.
Milnor number
The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
-
D.
Donaldson invariants
Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
quadratic form invariant ⓘ topological invariant ⓘ |
| appearsIn |
4-manifold topology
ⓘ
classification of quadratic forms over F2 ⓘ cobordism theory ⓘ theory of surface knots ⓘ |
| appliesTo | quadratic forms over fields of characteristic two ⓘ |
| computableFrom |
Alexander polynomial of a knot modulo 2
ⓘ
Seifert form of a knot ⓘ quadratic enhancement of the intersection pairing on a surface ⓘ |
| definedAs |
parity of the number of elements where the quadratic form takes value 1 in a suitable basis
ⓘ
sum over a symplectic basis of values of a quadratic form modulo 2 ⓘ |
| definedFor |
nondegenerate quadratic forms over F2
ⓘ
nondegenerate quadratic forms over finite fields of characteristic two ⓘ |
| fieldOfStudy |
algebraic topology
ⓘ
knot theory ⓘ quadratic form theory ⓘ surgery theory ⓘ |
| hasProperty |
additive under orthogonal sum of quadratic forms
ⓘ
gives a complete invariant of nonsingular quadratic forms over F2 up to isomorphism ⓘ is a concordance invariant of knots ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| introducedBy | Cahit Arf ⓘ |
| introducedInContext | classification of quadratic forms in characteristic two ⓘ |
| invariantUnder |
isometries of quadratic forms
ⓘ
knot concordance ⓘ stable equivalence of quadratic forms in characteristic two ⓘ |
| namedAfter | Cahit Arf ⓘ |
| relatedTo |
Brown invariant
ⓘ
Milnor invariants ⓘ Pin- structures ⓘ Rokhlin invariant ⓘ quadratic refinements of intersection forms ⓘ spin structures ⓘ |
| takesValue |
0
ⓘ
1 ⓘ |
| usedFor |
classification of nonsingular quadratic forms over fields of characteristic two
ⓘ
classifying even-dimensional manifolds with certain quadratic refinements ⓘ distinguishing knots up to concordance ⓘ studying framed manifolds in surgery theory ⓘ |
| usedInResult |
Kervaire–Milnor results on knot concordance
ⓘ
classification of nonsingular quadratic forms over fields of characteristic two ⓘ surgery obstructions in dimension four ⓘ |
| valueRange |
elements of Z/2Z
ⓘ
{0,1} ⓘ |
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: Arf invariant Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.