# Reverse use of Seifert-van Kampen Theorem?

I am trying to use S-vK Theorem in reverse; what I know are as follows:

1. $$U$$ and $$V$$ satisfy the requirements (open, path-connected), $$U\cup V = X$$, $$U \cap V = N$$
2. $$\pi_1(N) = \langle c,d| cd=dc\rangle$$
3. $$\pi_1(U) = ??$$
4. $$\pi_1(V) = \langle d\rangle$$
5. $$\pi_1(X) = \langle a,b|a^p=b^q\rangle$$, where $$p,q \in \mathbb{Z}$$
6. When $$c$$ and $$d$$ are injected into $$X$$, they become identity and $$b$$ respectively.

In fact does it work this way? Any other help? Thank you very much.

I know this is an old question, but here's an answer anyway: let $\pi_1(U) = \langle a,d | a^p = d^q \rangle$ and the map $f: \langle c,d | cd = dc \rangle \to \pi_1(U)$ be given by $f(c) = e, f(d) = d$. The pushout of $\pi_1(U)$ and $\pi_1(V)$ over $\pi_1(N)$ is isomorphic to the $\pi_1(X)$ above.
Alternatively $\pi_1(U) = \langle a,d,c | a^p = b^q, cd = dc \rangle$ and $f(c) = c, f(d) = d$. This gives the same pushout but is not isomorphic to the previous $\pi_1(U)$.