Gluing Karcher-Scherk saddle towers II: Singly periodic minimal surfaces

This is the second in a series of papers that construct minimal surfaces by gluing singly periodic Karcher--Scherk saddle towers along their wings. This paper aims to construct singly periodic minimal surfaces with Scherk ends. As in the first paper, we prescribe phase differences between saddle towers, and obtain many new examples without any horizontal reflection plane. This construction is not very different from previous ones, hence we will only provide sketched proofs. We will however study the embeddedness with great care. Previously, embeddedness can not be determined in the presence of "parallel" Scherk ends, as it was not clear if they bend towards or away from each other. In a recent study, the bending was completely ignored and embeddedness was falsely claimed. We correct this mistake by carefully analysing slight bendings, thus identify scenarios where the constructed surfaces are indeed embedded.