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

Abstract

In 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 [2]. These surfaces were introduced by Cooper [9,10] for 2-component links and were used to compute Alexander invariants, signatures and nullities. Cooper presents a sequence of geometric moves which relate any two C-complexes for a 2-component link. Cimasoni gives an extension of Cooper's moves, providing 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. Cimasoni's result is incorrect. We present counterexamples, a correction with detailed proof, and check that subsequent results relying on this lemma still hold.