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.

Definition by Seifert matrix

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.

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]

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

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

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.

Arf as knot concordance invariant

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.

Notes

1. Kauffman (1987) p.74
2. Kauffman (1987) pp.75–78
3. Robertello, Raymond, An Invariant of Knot Corbordism, Communications on Pure and Applied Mathematics, Volume 18, pp. 543–555, 1965
4. 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. 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.