Before revision:  These homotopy classes can be represented by orthogonal polylines in \Gamma_I . We argue that the structural difference between the two drawings can be captured by the spirality of the wires in \Gamma_I, which guides our morph from \Gamma_I to \Gamma_O.
After revision:  These homotopy classes can be represented by orthogonal polylines in \Gamma_I that share topological properties with the wires in \Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in \Gamma_I, which guides our morph from \Gamma_I to \Gamma_O.