Structure and classification of torus theta-curves

We study theta-curves embedded in a standard torus in the 3-sphere. We show that each nontrivial torus knot together with an essential arc determines a prime theta-curve, yielding explicit infinite families of prime theta-curves. We compute their constituent knots and identify the structure governing these embeddings, which leads to a complete classification of torus theta-curves up to ambient isotopy and homeomorphism of the 3-sphere. In particular, Kinoshita's theta-curve does not lie on a standard torus.