Minimal surfaces in the 3-sphere by desingularizing intersecting Clifford tori

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an additional Clifford torus equidistant from the original two circles of intersection, so that the latter torus orthogonally intersects each of the former $k$ tori along a pair of disjoint orthogonal circles, near which the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type resemble surfaces constructed by Choe and Soret \cite{CS} by different methods where the number of handles desingularizing each circle is the same. There is a plethora of new examples which are more complicated and on which the number of handles for the two circles differs. Examples of the second type are new as well.