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 to a Seifert surface. If F is a Seifert surface of a knot, then the homology group 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. The Arf invariant of this quadratic form is the Arf invariant of the knot. Definition by Seifert matrix Let 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. This means that V is a matrix with the property that is a symplectic matrix. 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 denotes the positive pushoff of a. Definition by pass equivalence This approach to the Arf invariant is due to Louis Kauffman. We define two knots to be pass equivalent if