how to calculate the fundamental group using van kampen theorem?

Question

i'm really stuck in calculating the fundamental group (pi1 ) of this surface... can someone please help me?.... i'm trying to use the van Kampen theorem but i don't know how to take the open sets here to work

Answer 1

As ArtW pointed out in the comments, the surface seems homeomorphic to a torus. This can be argued as follows:

• The knotted tunnel in the middle of the sphere never self-intersects or interacts with the sphere, so it is homeomorphic to a straight tunnel passing through the sphere
• A tunnel passing through a sphere is homeomorphic to a tunnel similarly connecting pole to pole but going outside the sphere instead of inside
• Now imagine deflating the sphere so it's the same width as the tunnel, giving a torus