Index of Bipolar Surfaces to Otsuki Tori

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