Arf Invariant

The Arf invariant is a Z/2Z valued concordance invariant. If the Alexander polynomial evaluated at -1 is equal to 3 or 5 mod 8, then the Arf invariant is 1, otherwise (if it is 1 or 7 mod 8) the Arf Invariant is 0. (Recall that the determinant of the knot is the absolute value of the Alexander polynomial evaluated at -1.)

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 1 if most classes have self-linking 1.