We develop theoretical aspects of refined Donaldson-Thomas theory for threefold flopping contractions, and use these to determine all DT invariants for infinite families of length 2 flops. Our results show that a refined version of the strong-rationality conjecture of Pandharipande--Thomas holds in this setting, and also that refined DT invariants do not determine flops. Our main innovation is the application of tilting theory to better understand the stability conditions and cyclic $A_\infty… CONTINUE READING