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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 H<sub>1</sub>(''F'', '''Z'''/2'''Z''') 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.
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| ==Definition by Seifert matrix==
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| Let <math>V = v_{i,j}</math> 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 (mathematics)|homology]] of the surface. This means that ''V'' is a 2''g'' × 2''g'' matrix with the property that ''V'' − ''V''<sup>T</sup> is a [[symplectic matrix]]. The ''Arf invariant'' of the knot is the residue of
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| :<math>\sum\limits^g_{i=1}v_{2i-1,2i-1}v_{2i,2i} \pmod 2.</math>
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| Specifically, if <math>\{a_i,b_i\}, i=1...g</math>, is a symplectic basis for the intersection form on the Seifert surface, then
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| :<math>Arf(K) = \sum\limits^g_{i=1}lk(a_i, a_i^{+})lk(b_i,b_i^{+}) \pmod 2.</math> | |
| where <math>a^i</math> denotes the positive pushoff of ''a''.
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| ==Definition by pass equivalence==
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| This approach to the Arf invariant is due to [[Louis Kauffman]].
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| We define two knots to be '''pass equivalent''' if they are related by a finite sequence of pass-moves,<ref>Kaufmann (1987) p.74</ref> which are illustrated below: (no figure right now)
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| Every knot is pass-equivalent to either the [[unknot]] or the [[Trefoil knot|trefoil]]; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.<ref>Kaufmann (1987) pp.75–78</ref>
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| 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.
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| ==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]].
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| ==Definition by Alexander polynomial==
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| This approach to the Arf invariant is by Raymond Robertello.<ref name="Robertello">Robertello, Raymond, [[Communications on Pure and Applied Mathematics]], Volume 18, pp. 543–555, 1965</ref> Let
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| : <math>\Delta(t) = c_0 + c_1 t + \cdots + c_n t^n + \cdots + c_0 t^{2n}</math>
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| be the Alexander polynomial of the knot. Then the Arf invariant is the residue of
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| : <math> c_{n-1} + c_{n-3} + \cdots + c_r</math>
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| modulo 2, where ''r'' = 0 for ''n'' odd, and ''r'' = 1 for ''n'' even.
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| Kunio Murasugi<ref name="Murasugi">Murasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69–72</ref> proved that the Arf invariant is zero if and only if Δ(−1) <math>\equiv</math> ±1 modulo 8.
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite book | last=Kauffman | first=Louis H. | authorlink = Louis Kauffman | title = Formal knot theory | year = 1983 | series=Mathematical notes | volume=30 | publisher=Princeton University Press | isbn = 0-691-08336-3 }}
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| * {{cite book | last = Kauffman | first= Louis H. | authorlink = Louis Kauffman | title = On knots | volume=115 | series=Annals of Mathematics Studies | publisher=Princeton University Press | year=1987 | isbn=0-691-08435-1 }}
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| * {{cite book | last=Kirby | first=Robion | authorlink = Robion Kirby | title = The topology of 4-manifolds | year = 1989 | series=Lecture Notes in Mathematics | volume=1374, | publisher =[[Springer-Verlag]] | isbn=0-387-51148-2 }}
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| {{Knot theory|state=collapsed}}
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| {{DEFAULTSORT:Arf invariant of a knot}}
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| [[Category:Knot invariants]]
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