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# 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(FZ/2Z) 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.

## 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 − VT 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 ${\displaystyle \{a_{i},b_{i}\},i=1...g}$, is a symplectic basis for the intersection form on the Seifert surface, then

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

where ${\displaystyle a^{+}}$ 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] 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

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

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) ${\displaystyle \equiv }$ ±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 ${\displaystyle \Delta (t)=p(t)p(t^{-1})}$ for some polynomial ${\displaystyle p(t)}$ with integer coefficients, we know that the determinant ${\displaystyle |\Delta (-1)|}$ of a slice knot is a square integer. As ${\displaystyle |\Delta (-1)|}$ 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, 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. 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.