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 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. The Arf invariant of this quadratic form is the Arf invariant of the knot.

Let ${\displaystyle V=v_{i,j}}$ 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

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

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

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

where lk is the link number and ${\displaystyle a^{+}}$ denotes the positive pushoff of a.

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]

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.

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.^[3] Let

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

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

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

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

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

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