In page 15 of the article [New Applications of Min-max Theory](https://arxiv.org/pdf/1409.7537.pdf), Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi\_{1,2}$ in the 3-sphere to have a Morse index 9, but there is no real evidence.


My question is, what suggests the index to be 9? Say, can we find a 9-parameter deformation of $\xi\_{1,2}$ which decreases its area?



Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?
Due to a recent preprint by Kapouleas and Wiygul (arXiv:1904.05812), the index of the Lawson $\xi\_{1,g}$ surface is $2g+3$. I have not looked at the paper in detail so far, but let me cite from the authors introduction:


"The ideas of our proof originate with work of NK on the approximate kernel for Scherk surfaces. Our approach requires a detailed understanding of the elementary geometry of $S^3$ and of the surfaces involved, especially their symmetries. The proof makes also heavy use of Alexandrov reflection in the style of Schoen’s. The Courant nodal theorem [ and an argument of Montiel- Ros play important roles as well. In ongoing work we hope to extend this result to determine the index and nullity of all Lawson surfaces desingularizing intersecting great two-spheres in the round three-sphere $S^3$."


