Index and nullity of minimal surface doublings, I

We prove that for any large enough $m\in \mathbb{N}$, the genus $\gamma =m+1$ equator-poles minimal surface doubling of the equatorial two-sphere ${\mathrm{\Sigma }}^0={\mathbb{S}}_{\mathrm{e}\mathrm{q}}^2$ in the round three-sphere ${\mathbb{S}}^3$, which has two catenoidal bridges at the poles and $m$ bridges equidistributed along the equatorial circle $\mathcal{C}$ of ${\mathrm{\Sigma }}^0$ and was discovered in earlier work of Kapouleas, has index $2\gamma +5=2m+7$ and nullity $6$, and so it has no exceptional Jacobi fields and is $C^1$-isolated.