Arf invariant (knot)

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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 imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.

1 Definition by Seifert matrix
2 Definition by pass equivalence
3 Definition by partition function
4 Definition by Alexander polynomial
5 References

[edit] Definition by Seifert matrix

Let $V = (v (i, j))$ be a Seifert matrix of the knot, constructed from a canonical set of curves on a Seifert surface of genus g. This means that $V$ is a $2g \times 2g$ matrix with the property that $V - V T$ is a symplectic matrix. The Arf invariant of the knot is the residue of

$\sum\limits^g_{i=1}v(2i-1,2i-1)v(2i,2i)$ modulo 2.

[edit] 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, 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.

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.

[edit] 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.

[edit] Definition by Alexander polynomial

This approach to the Arf invariant is by Raymond Robertello^[1]. Let $\Delta(t) = c_0 + c_1 t + \dots + c_n t^n + \dots + c_0 t^{2n}$ be the Alexander polynomial of the knot. Then the Arf invariant is the residue of $c_{n-1} + c_{n-3} + \dots + c_r$ modulo 2, where r = 0 for n odd, and r = 1 for n even.

Kunio Murasugi^[2] proved that the Arf invariant is zero if and only if $\Delta(-1) \equiv \pm 1$ modulo 8.

[edit] References

^ 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

Kauffman, Louis (1983). Formal knot theory. Mathematical notes, 30, Princeton University Press. ISBN 0-691-08336-3.
Kirby, Robion (1989). The topology of 4-manifolds. Lecture Notes in Mathematics, no. 1374, Springer-Verlag. ISBN 0-387-51148-2.