  In groundbreaking work from 2004, Cimasoni gave a geometric computation of
the multivariable Conway potential function in terms of a generalization of a
Seifert surface for a link called a C-complex. Lemma 3 of that paper provides a
family of moves which relates any two C-complexes for a fixed link. This allows
for an approach to studying links from the point of view of C-complexes and in
following papers it has been used to derive invariants. This lemma is false. We
present counterexamples, a correction with detailed proof, and an analysis of
the consequences of this error on subsequent works that rely on this lemma.
