Knot Arf invariant from the Seifert form

Assume full AC. Let $K$ be a smooth oriented knot and $F$ a connected compact smooth oriented spanning surface of genus $g$. The Seifert matrix $V$ defines a function on $H_1(F;{\mathbb{F}}_2)$ by $q_F([x])=\mathrm{lk}(x^{+},x)(\mathrm{m}\mathrm{o}\mathrm{d}2)=x^{T}Vx(\mathrm{m}\mathrm{o}\mathrm{d}2),$ where an integral lift of the mod-two homology class is used in the last expression. Its polarization is the mod-two intersection form. A symplectic basis is an ordered basis $(a_1,b_1,\dots ,a_g,b_g)$ with $a_i\cdot b_j={\delta }_{ij}$ and all other pairings zero. Define $\mathrm{Arf}(q_F)={\sum }_{i=1}^g{q}_F(a_i)q_F(b_i)(\mathrm{m}\mathrm{o}\mathrm{d}2)\mathrm{.}$ The quadratic-enhancement lemma proves that this value is independent of symplectic basis; the S-equivalence and Conway bridge proves independence of $F$, so the resulting knot invariant is denoted $\mathrm{Arf}(K)$. For $g=0$ the empty sum is $0$.

The positive-push Seifert pairing and its matrix

The skew part is the integral intersection form

Well-definedness and basis independence of the quadratic enhancement ↗

Independence of the spanning surface ↗