Skew loops in flat tori

We produce skew loops—loops having no pair of parallel tangent lines—homotopic to any loop in a flat torus or other quotient of R ⁿ . The interesting case here is n = 3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop as tangent indicatrix. A fellowship from the Lady Davis foundation helped support this work.