Arf Invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, of the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.

In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to Dickson (1901), even for any finite field of characteristic 2, and it follows from Arf's results for an arbitrary perfect field. An assessment of Arf's results in the framework of the theory of quadratic forms can be found in,

The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of (4k+2)-dimensional manifolds (singly even-dimension manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4k-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.

Read more about Arf Invariant:  Definitions, Arf's Main Results, Quadratic Forms Over F2, The Arf Invariant in Topology

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... For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant ... The skew-symmetric form is an invariant of the surface Σ, whereas the skew-quadratic form is an invariant of the embedding Σ ⊂ R3, e.g ... The Arf invariant of the skew-quadratic form is a framed cobordism invariant generating the first stable homotopy group ...
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... The corresponding quadratic form (see below) has Arf invariant 1 ... The corresponding quadratic form (see below) has Arf invariant 0 ... Then the Arf invariant of this quadratic form can be used to distinguish the two extra special groups ...
Arf Invariant Of A Knot
... In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated ... The Arf invariant of this quadratic form is the Arf invariant of the knot ...
The Arf Invariant in Topology - Examples
... The Arf invariant of the framed surface is now defined Note that, so we had to stabilise, taking to be at least 4, in order to get an element of ... The Arf invariant of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing ... putting a spin structure on our surface.) Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed cobordism group, which is generated by the ...
Arf Invariant Of A Knot - Definition By Alexander Polynomial
... This approach to the Arf invariant is by Raymond Robertello ... Then the Arf invariant is the residue of modulo 2, where r = 0 for n odd, and r = 1 for n even ... Kunio Murasugi proved that the Arf invariant is zero if and only if Δ(−1) ±1 modulo 8 ...