# how to calculate the fundamental group using van kampen theorem?

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

• If I interprete your drawing well, this is homeomorphic to a torus, so no need to use Van Kampen
– ArtW
Commented Apr 20, 2018 at 14:00
• torus? i can't understand how? Commented Apr 20, 2018 at 14:02
• the knotted tube in the middel is just homeomorphic to a straight tube
– ArtW
Commented Apr 20, 2018 at 14:04
• @ArtW : Thank you. Commented Apr 20, 2018 at 14:06
• but it is clinging from both sides to the sphare... sorry i can't understand how is it isomorphic to a straght line, if i dont consider the both sides it's tru but now... Commented Apr 20, 2018 at 14:07

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