Disjoint circuits in the cartesian product of two directed cycles

We show that the Cartesian product of two directed cycles Z _a X Z _b has r disjointly embedded circuits C₁, C₂, ?, C_r with specified knot classes knot(C_i) = (m_i, n_i), for i = 1, 2, ?, r, if and only if there exist relatively prime non-negative integers m and n such that knot(C_i) = (m, n), for i = 1, 2, ?, r, and r(am + bn) ≦ ab. We generalize this result to the Cayley digraph on a finite abelian group with a two-element generating set.