Follow up on cinquefoil knot Using the following Seifert surface of the cinquefoil knot !enter image description here I get the following Seifert matrix (of linking numbers): $$ S = \begin{pmatrix}- 1 &1 &0 &0 \\\ 0 &-1 &1& 0 \\\ 0& 0 &-1& 1\\\ 0 &0 &0& -1\end{pmatrix}$$ I compute the matrix of the corresponding bilinear form $$ I = S^T - S = \begin{pmatrix} 0 & -1 & 0 & 0 \\\ 1 & 0 & -1 & 0 \\\ 0 & 1 & 0 & -1 \\\ 0 & 0 & 1 & 0 \end{pmatrix}$$ and a corresponding symplectic basis $$e_1 = \begin{pmatrix}0 \\\0 \\\0 \\\1 \end{pmatrix}$$ $$f_1 = \begin{pmatrix} 0 \\\1 \\\1 \\\0\end{pmatrix}$$ $$f_2 = \begin{pmatrix}1 \\\0 \\\0 \\\0\end{pmatrix}$$ $$e_2 = \begin{pmatrix}1 \\\1 \\\1 \\\1\end{pmatrix}$$ And I get $\mathrm{Arf}(K) = e_1^T S e_1 f_1^T S f_1 + e_2^T S e_2 f_2^T S f_2 = 2 \equiv_2 0 \neq 1$ Where is my mistake? Thanks for your help.

For the matrices $I$ and $S$ the following is a symplectic basis

$$ e_1 = \begin{pmatrix} 1 \\\ -1 \\\ 1 \\\ 0 \end{pmatrix}$$

$$ f_1 = \begin{pmatrix} 0 \\\ 1 \\\ 1 \\\ 0 \end{pmatrix}$$

$$e_2 = \begin{pmatrix} -1 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}$$ $$f_2 = \begin{pmatrix} -1 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$

yielding

$$ \mathrm{Arf}(K) = e_1^T S e_1 f_1^T S f_1 + e_2^T S e_2 f_2^T S f_2 = 9 \equiv_2 1$$ as it should.