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

TL;DR

This paper identifies an error in Cimasoni's 2004 work on C-complexes, provides counterexamples, corrects the lemma, and discusses the impact on related link invariants and subsequent research.

Contribution

It corrects a fundamental lemma about moves relating C-complexes, clarifying the implications for geometric link invariants derived from Cimasoni's approach.

Findings

Counterexamples show the original lemma is false.

A corrected version of the lemma is provided with detailed proof.

The correction impacts subsequent research relying on the original lemma.

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. 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.