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{{short description|Knot invariant named after Cahit Arf}}
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]] {{nowrap|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 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==

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 {{nowrap|2''g'' × 2''g''}} matrix with the property that {{nowrap|''V'' − ''V''<sup>T</sup>}} is a [[symplectic matrix]]. The ''Arf invariant'' of the knot is the residue of 

:<math>\sum\limits^g_{i=1} v_{2i-1,2i-1} v_{2i,2i} \pmod 2.</math>

Specifically, if <math>\{a_i, b_i\}, i = 1 \ldots g</math>, is a symplectic basis for the intersection form on the Seifert surface, then 

:<math>\operatorname{Arf}(K) = \sum\limits^g_{i=1}\operatorname{lk}\left(a_i, a_i^+\right)\operatorname{lk}\left(b_i, b_i^+\right) \pmod 2.</math>
where lk is the [[link number]] and <math>a^+</math> 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.<ref>Kauffman (1987) p.74</ref>

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>Kauffman (1987) pp.75–78</ref>

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 an [[Ising model]] on a [[knot diagram]].<ref>{{citation
 | last = Jones | first = Vaughan F. R.
 | doi = 10.1038/scientificamerican1190-98
 | issue = 5
 | journal = Scientific American
 | jstor = 24996978
 | mr = 1079724
 | pages = 4, 98–103
 | title = Knot theory and statistical mechanics
 | volume = 263
 | year = 1990}}</ref>

==Definition by Alexander polynomial==
This approach to the Arf invariant is by Raymond Robertello.<ref name="Robertello">Robertello, Raymond, An Invariant of Knot Corbordism, [[Communications on Pure and Applied Mathematics]], Volume 18, pp. 543&ndash;555, 1965</ref>  Let

: <math>\Delta(t) = c_0 + c_1 t + \cdots + c_n t^n + \cdots + c_0 t^{2n}</math>

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

: <math> c_{n-1} + c_{n-3} + \cdots + c_r</math>

modulo 2, where {{nowrap|''r'' {{=}} 0}} for ''n'' odd, and {{nowrap|''r'' {{=}} 1}} for ''n'' even.

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&ndash;72</ref> proved that the Arf invariant is zero if and only if {{nowrap|Δ(−1) ≡ ±1 modulo 8}}.

==Arf as knot concordance invariant==

From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a [[slice knot]] <math> K \subset \mathbb{S}^3 </math> factors as <math> \Delta(t) = p(t) p\left(t^{-1}\right) </math> for some polynomial <math> p(t) </math> with integer coefficients, we know that the determinant <math> \left| \Delta(-1) \right| </math> of a slice knot is a square integer. As <math> \left| \Delta(-1) \right| </math> 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==
<references/>

==References==
* {{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 }}
* {{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 }}
* {{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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[[Category:Knot invariants]]