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 H_{1}(*F*, **Z**/2**Z**) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.

## Contents

## 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 2*g* × 2*g* matrix with the property that *V* − *V*^{T} 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 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 they are related by a finite sequence of pass-moves,^{[1]} which are illustrated below: (no figure right now)

Every knot is pass-equivalent to either the unknot or the trefoil; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.^{[2]}

Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.

## Definition by partition function

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.

## Definition by Alexander polynomial

This approach to the Arf invariant is by Raymond Robertello.^{[3]} Let

be the Alexander polynomial of the knot. 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.

## Arf as knot concordance invariant

From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a slice knot factors as for some polynomial with integer coefficients, we know that the determinant of a slice knot is a square integer. As is an odd integer, it has to be congruent to 1 modulo 8. Combined with Murasugi's result this shows that the Arf invariant of a slice knot vanishes.

## Notes

**^**Kauffman (1987) p.74**^**Kauffman (1987) pp.75–78**^**Robertello, Raymond, Communications on Pure and Applied Mathematics, Volume 18, pp. 543–555, 1965**^**Murasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69–72

## References

- Kauffman, Louis H. (1983).
*Formal knot theory*. Mathematical notes.**30**. Princeton University Press. ISBN 0-691-08336-3. - Kauffman, Louis H. (1987).
*On knots*. Annals of Mathematics Studies.**115**. Princeton University Press. ISBN 0-691-08435-1. - Kirby, Robion (1989).
*The topology of 4-manifolds*. Lecture Notes in Mathematics. 1374,. Springer-Verlag. ISBN 0-387-51148-2.