# Wilson loop expectation value in $RP^3$ using Dehn surgery

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:

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.