  Motivated by the problem of determining unknotted routes for the scaffolding
strand in DNA origami self-assembly, we examine existence and knottedness of
A-trails in graphs embedded on the torus. We show that any A-trail in a
checkerboard-colorable torus graph is unknotted and characterize the existence
of A-trails in checkerboard-colorable torus graphs in terms of pairs of
quasitrees in associated embeddings. Surface meshes are frequent targets for
DNA nanostructure self-assembly, and so we study both triangular and
rectangular torus grids. We show that, aside from one exceptional family, a
triangular torus grid contains an A-trail if and only if it has an odd number
of vertices, and that such an A-trail is necessarily unknotted. On the other
hand, while every rectangular torus grid contains an unknotted A-trail, we also
show that any torus knot can be realized as an A-trail in some rectangular
grid. Lastly, we use a gluing operation to construct infinite families of
triangular and rectangular grids containing unknotted A-trails on surfaces of
arbitrary genus. We also give infinite families of triangular grids containing
no unknotted A-trail on surfaces of arbitrary nonzero genus.
