Arf invariant of a 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 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

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Let   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 VVT is a symplectic matrix. The Arf invariant of the knot is the residue of

 

Specifically, if  , is a symplectic basis for the intersection form on the Seifert surface, then

 

where lk is the link number and   denotes the positive pushoff of a.

Definition by pass equivalence

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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

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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

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This approach to the Arf invariant is by Raymond Robertello.[3] Let

 

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

 

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

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From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a slice knot   factors as   for some polynomial   with integer coefficients, we know that the determinant   of a slice knot is a square integer. As   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

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  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

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  • 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.