I am currently reading Quantum Invariants of Knots and 3-manifolds by G.Turaev. In Section 2.2, the book gives a way to define invariant of 3-manifolds by Dehn surgery and signature.
I have known what is the signature of 4-manifold by Poincaré-Lefschetz duality. But I meet some difficulty in computing concrete cases. For example, let $W$ be a 4-manifold with boundary $L(p,q)$. How to compute the signature $\sigma(W)$ of $W$? I hear that it is exactly the signature of link matrix of $L$, where $L$ is a framed link such that we get $L(p,q)$ via Dehn surgery in $S^3$. In detail, let $p/q = a_1-1/(a_2-1/(a_3-\dots)\dots)$, $a_i\in\mathbb{Z}$. Then the link matrix is $$ lk(L) = \left[ \begin{matrix} a_1 & 1 & 0 & \dots & 0\\ 1 & a_2 & 1 & \dots & 0\\ 0 & 1 & a_3 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 &\dots & 1 \end{matrix} \right] $$
Maybe there is $\sigma(M)=\sigma(lk(L))$. Is this true? If it is true, could you please give a topological explaination? Thank you!