For n=2 you do have a product link. You can see it by writing the pages $(S^1\times I)$ of the open book collectively as $(S^1 \times I) \times S^1$, and then gluing in two solid tori $S^1\times D^2$ by using the disks $\{*\} \times D^2$ to cap off the annuli $(\{*\} \times I) \times S^1$ (which consist of one arc from each page). Thus in the surgered manifold the annuli $(\{*\}\times I)\times S^1$ can be closed up to form spheres $\{*\}\times S^2$, one for each point of $S^1$, and the link $L_2$ contributes a pair of points to each of those spheres.

The argument only works for $n>0$ because when $n=0$ it isn't really an open book decomposition anymore, and then you really do have $S^2\times S^1$. You can check the n=2 case directly by Kirby calculus, though: it's 0-surgery on an unknot and on two of its meridians, and if you blow up one of those meridians you get a chain of unknots with surgery coefficients -1,-1,0,0. Blow down the second to get a chain with coefficients 0,1,0, and then the middle one, and then either of the remaining ones and you're left with a single 0-framed unknot, which gives $S^1\times S^2$.

@Vivek: it's in section 9.2.2 of "Introduction to Vassiliev Knot Invariants" by Chmutov, Duzhin, and Mostovoy, available at pdmi.ras.ru/~duzhin/papers/cdbook/cdbook.pdf.

@minimax: see katlas.org/wiki/Cabling for an example. You have to import the program CableComponent.m, and then I believe you want to use CableComponent[BR[TorusKnot[19,3]], Knot[3,1]]. (I haven't used CableComponent before, but this knot seems to at least have the right Alexander polynomial according to the formula $\Delta(t) = \Delta_{T_{3,19}}(t) * \Delta_{T_{2,3}}(t^3)$.)

The KnotTheory` Mathematica package, available with lots of documentation and examples at katlas.org/wiki/Setup, can compute HOMFLY and many other knot polynomials and even produce cables for you automatically.