We consider invariants of mapping tori of symplectomorphisms of a symplectic surface S, such as symplectic field theory, contact homology, and periodic Floer homology, for the standard stable Hamiltonian structure on the mapping torus. These invariants involve counts of holomorphic curves in R times the mapping torus. We obtain a number theoretic description of all rigid holomorphic curves in the case S = T2, and obtain various pair-of-pants invariants for symplectomorphisms on higher genus surfaces. Our method involves reinterpreting counts of holomorphic pairs of pants in R times the mapping torus as counts of index -1 triangles between Lagrangians in S x S for certain 1-parameter families of almost-complex structures.