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I am currently reading Guadagnini's The link invariants of Chern-Simons field theory, the part where he computes some examples of expectation values for different spaces.

For $S^2 \times S^1$, he uses one unknot with surgery coefficient $r=0$ to perform Dehn surgery on $S^3$. His computation is the following: S2 x S1 computation

I can see where all the expressions come from from previous chapters, however I am unable to see where the fundamental group is used (since he computes it right before this) or where the value of $r$ is used. The next computation, now for $RP^3$ is the same unknot but with a surgery coefficient $r=2$. I am not able to see how this value comes into play.

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