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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

enter image description here

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  • $\begingroup$ If I interprete your drawing well, this is homeomorphic to a torus, so no need to use Van Kampen $\endgroup$ – ArtW Apr 20 '18 at 14:00
  • $\begingroup$ torus? i can't understand how? $\endgroup$ – user297564 Apr 20 '18 at 14:02
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    $\begingroup$ the knotted tube in the middel is just homeomorphic to a straight tube $\endgroup$ – ArtW Apr 20 '18 at 14:04
  • $\begingroup$ @ArtW : Thank you. $\endgroup$ – HK Lee Apr 20 '18 at 14:06
  • $\begingroup$ 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... $\endgroup$ – user297564 Apr 20 '18 at 14:07
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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
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