  An open question akin to the slice-ribbon conjecture asks whether every
ribbon knot can be represented as a symmetric union. Next to this basic
existence question sits the question of uniqueness of such representations.
Eisermann and Lamm investigated the latter question by introducing a notion of
symmetric equivalence among symmetric union diagrams and showing that
inequivalent diagrams can be detected using a refined version of the Jones
polynomial. We prove that every topological spin model gives rise to many
effective invariants of symmetric equivalence, which can be used to distinguish
infinitely many symmetric union diagrams representing the same link. We also
show that such invariants are distinct from the refined Jones polynomial and we
use them to provide a partial answer to a question left open by Eisermann and
Lamm.
