Arf invariant (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 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 $V=(v(i,j))$ 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 $2g\; imes\; 2g$ matrix with the property that $V\; -\; V^T$ is a symplectic matrix. The "Arf invariant" of the knot is the residue of

:$sumlimits^g\_\{i=1\}v(2i-1,2i-1)v(2i,2i)$ modulo 2.

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.

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 RobertelloRobertello, Raymond, Communications on Pure and Applied Mathematics, Volume 18, pp. 543-555, 1965] . 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 MurasugiMurasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69-72] proved that the Arf invariant is zero if and only if $Delta(-1)\; equiv\; pm\; 1$ modulo 8.

References

* cite book
author = Kauffman, Louis
authorlink = Louis Kauffman
title = Formal knot theory
year = 1983
publisher = Mathematical notes, 30, Princeton University Press
id = ISBN 0-691-08336-3
* cite book
author = Kirby, Robion
authorlink = Robion Kirby
title = The topology of 4-manifolds
year = 1989
publisher = Lecture Notes in Mathematics, no. 1374, Springer-Verlag
id = ISBN 0-387-51148-2