Arf-Robertello Invariant

The Arf invariant of a knot 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.

For proper links (all linking numbers even) there is the Robertello invariant. For an n-component proper link, use (n-1) bands to form a knot. Robertello proved that the arf invariant of the resulting knot depends only on the initial link. This invariant obstructs the initial link from being (topologically or smoothly) slice.


[1] Robertello, R. "An invariant of knot cobordism," Comm. Pure Appl. Math. 18 (1965), 543-555.