In page 15 of the article New Applications of Min-max Theory, 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?


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