New low-genus desingularizations of three Clifford tori and related characterizations

For each nonnegative integer $m$ we construct in the round three-sphere a closed embedded minimal surface of genus $48m+25$ which can be interpreted as a desingularization of the union of three Clifford tori intersecting pairwise orthogonally, along a total of six great circles. Each such surface is generated, under the action of a group of symmetries, by a disc with hexagonal boundary, all of whose sides are contained in great circles. We prove a uniqueness result for this disc, and, as a corollary, we characterize these surfaces. This characterization implies that similar surfaces we constructed for sufficiently high $m$ by gluing methods, in an earlier article, coincide with the ones here. For low $m$ the surfaces constructed here are new. Similarly, we prove uniqueness of the generating discs for one of two families constructed by Choe and Soret (namely the surfaces they call odd) and show that these surfaces also coincide, when of sufficiently high genus, with surfaces we have constructed by gluing.