Hopf Tori

Willmore tori in the 3-sphere

The Hopf tori are equivariant Willmore tori with torus knot number 1:1. They were found by Ulrich Pinkall in 1985 [1].

3-lobed Hopf torus
4-lobed Hopf torus
5-lobed Hopf torus
6-lobed Hopf torus
7-lobed Hopf torus
8-lobed Hopf torus

References

  1. Dirk Ferus and Franz Pedit, S1-equivariant minimal tori in S4 and S1-equivariant Willmore tori in S3, Math. Z. 204 (1990), no. 2, 269–282.
  2. U. Pinkall, Hopf tori in S3, Invent. Math. 81 (1985), no. 2, 379–386.