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

Abstract

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 [2]. 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.

An algorithm to calculate generalized Seifert matrices

“…These 2 µ generalized Seifert matrices can be used to define and calculate the Conway potential function, which in turn determines the multivariable Alexander polynomial [Ci04,p. 128] [DMO21]. We recall the formulae in Theorem 1.…”