Moves relating C-complexes: A correction to Cimasoni's “A geometric construction of the Conway potential function”

In work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms generalization Seifert surface for link called C-complex [2]. These surfaces were introduced by Cooper [9], [10] 2-component links and used to compute Alexander invariants, signatures nullities. presents sequence moves which relate any two C-complexes link. gives an extension Cooper's moves, providing family relates fixed This allows approach studying point view following papers it has been derive invariants. Cimasoni's result is incorrect. We present counterexamples, correction with detailed proof, check that subsequent results relying on this lemma still hold.