I have troubles understanding the linking form and need some help with it.
What I understand so far is the following:
For some $2n+1$ dimensional rational homology sphere $M$ we can consider the composition $\varphi$ of the following maps:
$$H_n(M;\mathbb{Z})\overset{PD}{\rightarrow}H_n(M;\mathbb{Z})H^{n+1}(M;\mathbb{Z})\overset{Bockstein^{-1}}{\rightarrow}H^n(M;\mathbb{Q}/\mathbb{Z})\overset{eval.mor.}{\rightarrow}Hom_\mathbb{Z}(H_n(M;\mathbb{Z}),\mathbb{Q}/\mathbb{Z})$$
Now the linking form on $M$ is defined to be the bilinear map $\lambda_M:H_n(M;\mathbb{Z})\times H_n(M;\mathbb{Z})\rightarrow \mathbb{Q}/\mathbb{Z}$ sending $(a,b)$ to $(\varphi(a))(b)$
Now for some knot $K$ we can consider the two-fold branched covering $\Sigma(K)$ of $S^3$ branched along $K$. This seems to be a rational homology sphere allowing us to consider the linking form on it which is a bilinear form $\lambda_M:H_1(\Sigma(K);\mathbb{Z})\times H_1(\Sigma(K);\mathbb{Z})\rightarrow \mathbb{Q}/\mathbb{Z}$
I am interested in proofing the fact that this form is isometric to the form
$$\mathbb{Z}^{2g}/(A+A^T)\mathbb{Z}^{2g}\times \mathbb{Z}^{2g}/(A+A^T)\mathbb{Z}^{2g}\rightarrow\mathbb{Q}/\mathbb{Z}\text{ mapping }([v],[w])\text{ to }v^T(A+A^T)^{-1}w$$ where $A$ is some Seifert-matrix for $K$ arising from some Seifert-surface of genus $g$. I don't even understand this statement fully yet as not every Seifert-matrix arises from some Seifert surface and I kind of want to have this isometry of forms for all Seifert-matrices and $2g$ just beeing their size in that case.
Any help or advice in this task or any further illuminations on the linking form would be greatly appreciated. Thanks in advance!