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?

Solution 1:

Mathematica gives as KnotData[]: KnotData[{5, 1}, "SeifertMatrix"]

{{-1, -1, 0, -1}, {0, -1, 0, 0}, {-1, -1, -1, -1}, {0, -1, 0, -1}}

Which is closer to the first "given" matrix. The KnotData gives the arf invariant for that knot:

KnotData[{5, 1}, "ArfInvariant"]

1

I know that only gives Mathematica as the authority for the answers but that is the best I can find.