Mathematical Physics: Classical Mechanics
《Mathematical Physics: Classical Mechanics》 · 第 104–105 页 · Chapter 10，Definitions 10.2、10.3 及链环二次型构造（不是一般代数二次型义）
The Arf invariant \(c(\psi)\) of the non-singular quadratic form \(\psi:V\to\mathbb Z/2\mathbb Z\) is the value, 0 or 1, taken more often by \(\psi(u)\) as \(u\) varies over the \(2^{2n}\) elements of \(V\). […] Let \(L\) be an oriented link in \(S^3\) with Seifert surface \(F\), the orientation being needed to define \(F\). Define \(q:H_1(F;\mathbb Z/2\mathbb Z)\to\mathbb Z/2\mathbb Z\) by \(q(x)=\alpha_2(x,x)\in\mathbb Z/2\mathbb Z\), where \(\alpha_2\) is the Seifert form \(\alpha\) (see Chapter 6) reduced modulo 2. […] Suppose that \(L\) has components \(\{L_i\}\) and that \(L\) has the property that \((*)\quad\operatorname{lk}(L_i,L-L_i)\equiv0\text{ modulo }2\). […] A second glance shows that \(\mathcal F\) induces a non-singular form on the quotient \(H_1(F;\mathbb Z/2\mathbb Z)/\iota_*H_1(\partial F;\mathbb Z/2\mathbb Z)\), where \(\iota\) is the inclusion map. […] Hence \(q\) induces a well-defined non-singular quadratic form \(\bar q:H_1(F;\mathbb Z/2\mathbb Z)/\iota_*H_1(\partial F;\mathbb Z/2\mathbb Z)\to\mathbb Z/2\mathbb Z\). The Arf invariant \(\mathcal A(L)\) of an oriented link \(L\) having the property \((*)\) is \(c(\bar q)\), where \(\bar q:H_1(F;\mathbb Z/2\mathbb Z)/\iota_*H_1(\partial F;\mathbb Z/2\mathbb Z)\to\mathbb Z/2\mathbb Z\) is the quadratic form described above.