Given $n$ points $p_1,\dots,p_n$ in $S^2$ one gets a product link $L_n=\{p_1,\dots,p_n\}\times S^1$ inside the closed 3-manifold $S^2\times S^1$, which can be looked at as a trivially framed link (by picking a tangent vector $v_i$ at each $p_i$ and dragging it along the factor $S^1$ to get the framing). I'm interested in the result of the surgery of $S^2\times S^1$ along $L_n$. Is it diffeomorphic to some ``space with a name'' (e.g., $S^3$, $S^1\times S^1\times S^1$, a lens space, etc?)

I'm interested in the answer for an explict computation of a Reshetikhin-Turaev invariant I'm faced with. Yet, the problem is purely topological and I expect there is well known answer, but I haven't been able to find it or work it out myself so far.