If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group.
be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surface of genus g which represent a basis for the first homology of the surface.
The Arf invariant of the knot is the residue of Specifically, if
, is a symplectic basis for the intersection form on the Seifert surface, then where lk is the link number and
This approach to the Arf invariant is due to Louis Kauffman.
We define two knots to be pass equivalent if they are related by a finite sequence of pass-moves.
Vaughan Jones showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a knot diagram.
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[4] proved that the Arf invariant is zero if and only if Δ(−1) ≡ ±1 modulo 8.
From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a slice knot
Combined with Murasugi's result, this shows that the Arf invariant of a slice knot vanishes.