The Arf Invariant


The Arf invariant is a Z/2Z valued concordance invariant. If the Alexander polynomial evaluated at 1 (eg. the determinant of the knot, up to sign) is 3 or 5 mod 8, the Arf invariant is 1, otherwise (if it is 1 or 7 mod 8) the Arf Invariant is 0.

The Arf invariant can be defined in terms of the classical Arf invariant of the Seifert form, reduced mod 2. That is, there is a quadratic form on the Z/2Z homology of a Seifert surface, given by computing the linking of a class and its push-off. The Arf invariant is 0 if a majority of classes have self-linking 0, and is 1 if most classes have self-linking 1.