<s>[INST] Seifert matrices and Arf invariant -- Cinquefoil knot I have computed the following Seifert matrix for the Cinquefoil knot:

$$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\ 
        0 & 1 & 1 & 0 \\ 0 &  1& 1 & 1\end{pmatrix}$$
I also found a Seifert matrix for this knot on the internet but I still don't know how to verify the correctness of a Seifert matrix. Hence:
Question 1: How can I verify that a Seifert matrix that I computed is correct?
Now, more importantly: 
I thought I could compute the Arf invariant as follows (where $K$ is my knot): 
$$ A(K) = A(q) = q(x_1) q(x_2) + q(x_3) q(x_4) = 1 \cdot 1 + 1 \cdot 1 = 0$$
where $x_1 = \begin{pmatrix}1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, x_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, x_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} , x_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$ and $q(x) = x^T S x$.
But this must be wrong for the following reasons: 1. Assuming the Seifert matrix I found on the internet is correct, I get the same Arf invariant. 2. The Arf invariant of the cinquefoil knot is $1$. 
3. "The Arf invariant is $0$ if a majority of classes have self-linking $0$, and is $1$ if most classes have self-linking $1$.". (Excerpt taken from here). The self-linking numbers are the diagonal entries of the Seifert matrix as far as I understand.)
Question 2: What is the correct way to compute the Arf invariant of a knot from a Seifert form / Seifert matrix?
 [/INST] What makes you think that $x_i$ form a symplectic basis for $S$? In fact they don't. Also, you don't want a symplectic basis for $S$ but rather for $I = S^T - S$. 
Your Seifert matrix must be wrong since if I compute the Arf invariant using the $S$ with a symplectic basis 
$$ e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\1 \end{pmatrix}
 e_2 = \begin{pmatrix} -1 \\ 1 \\ 0 \\1 \end{pmatrix} 
  f_1 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}
 f_2 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}$$  I get zero when I should be getting $1$.
Using the Seifert matrix you linked, 
$$ S = 
\begin{pmatrix} -1 & -1 & 0 & -1 \\ 0 & -1 & 0 & 0 \\ -1 & -1 & -1 & -1 \\ 0 & -1 & 0 & -1 \end{pmatrix}$$
and a corresponding symplectic basis for
$$ I = S^T - S = \begin{pmatrix} 0 & 1 & -1 & -1 \\
-1 & 0 & -1 & -1 \\ 1 & 1 & 0 & 1 \\ -1 & 1 & -1 & 0\end{pmatrix}$$
$$ e_1 = \begin{pmatrix} -1 \\ 0 \\ 0 \\1 \end{pmatrix}
 e_2 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} 
  f_1 = \begin{pmatrix} -1 \\ 0 \\ 0 \\1 \end{pmatrix}
 f_2 = \begin{pmatrix} -1 \\ 0 \\ 1 \\0 \end{pmatrix}$$
yields
$$ A(K) = e_1^T S e_1 e_2^T S e_2 + f_1^T S f_1 f_2^T S f_2 = 7 \equiv_2 1$$
as expected.
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