Index of bipolar surfaces to Otsuki tori

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