Arf invariant

E654172

algebraic invariant quadratic form invariant topological 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.

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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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
relatedTo	Brown invariant NERFINISHED ⓘ Milnor invariants NERFINISHED ⓘ Pin- structures ⓘ Rokhlin invariant NERFINISHED ⓘ 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 NERFINISHED ⓘ classification of nonsingular quadratic forms over fields of characteristic two ⓘ surgery obstructions in dimension four ⓘ
valueRange	elements of Z/2Z ⓘ {0,1} ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cahit Arf → knownFor → Arf invariant ⓘ

Cahit Arf → notableConcept → Arf invariant ⓘ