# fundamental group of two circles which intersect at two common points

I am trying to solve this by Van-Kampen's theorem. What I do is just move a point at left side and a point at right side. Then I get two open sets whose intersection deformation retracts to a circle. Each open set deformation retracts to a figure 8.

So what is the next? Just free product with 4 basis module the normal subgroup generated by the fundamental group of a circle? I can not see what is this normal subgroup. Is it just Z?

This is how I pictured the described space. It's easy to give a CW decomposition where the red arc above is a cell. It's contractible, so this CW complex is homotopically equivalent to what you get after contracting that red arc to a point (see Hatcher's Algebraic Topology, Proposition 0.17, pages 15-16); and that's three circles wedged at a point, whose fundamental group is simply $\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$.