Index of bipolar surfaces to Otsuki tori

For each rational number $p\mathrm{/}q\in (1\mathrm{/}2,\sqrt{2}\mathrm{/}2)$ one can construct an ${\mathbb{S}}^1$-equivariant minimal torus in ${\mathbb{S}}^3$ called Otsuki torus and denoted by $O_{p\mathrm{/}q}$. The Lawson’s bipolar surface construction applied to $O_{p\mathrm{/}q}$ gives a minimal torus ${\tilde{O}}_{p\mathrm{/}q}$ in ${\mathbb{S}}^4$. In this paper we give upper and lower bounds on the Morse index and the nullity of these tori for $p\mathrm{/}q$ close to $\sqrt{2}\mathrm{/}2$. We also state a numerically assisted conjecture concerning the general case.