Arf invariant of a knot
# 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 $H_1(F,$ 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. The Arf invariant of this quadratic form is the Arf invariant of the knot.

## Definition by Seifert matrix

Let $V=v_{i,j}$ 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 2g × 2g matrix with the property that V − $V^T$ is a symplectic matrix. The Arf invariant of the knot is the residue of

    $\sum \limits _{i=1}^{g}v_{2i-1,2i-1}v_{2i,2i}{\pmod {2}}.$

Specifically, if $\{a_{i},b_{i}\},i=1\ldots g$, is a symplectic basis for the intersection form on the Seifert surface, then

    $\operatorname {Arf} (K)=\sum \limits _{i=1}^{g}\operatorname {lk} \left(a_{i},a_{i}^{+}\right)\operatorname {lk} \left(b_{i},b_{i}^{+}\right){\pmod {2}}.$

where lk is the link number and $a^{+}$ 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.

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.

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. Let

    $\Delta (t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}$

be the Alexander polynomial of the knot. Then the Arf invariant is the residue of

    $c_{n-1}+c_{n-3}+\cdots +c_{r}$

modulo 2, where r = 0 for n odd, and r = 1 for n even.

Kunio Murasugi 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 $K\subset \mathbb {S} ^{3}$ factors as $\Delta (t)=p(t)p\left(t^{-1}\right)$ for some polynomial $p(t)$ with integer coefficients, we know that the determinant $\left|\Delta (-1)\right|$ of a slice knot is a square integer. As $\left|\Delta (-1)\right|$ 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.

