Remark on twists of Frobenius algebra and link homology

TL;DR

This paper revisits the relationship between Frobenius algebra twists and link homology, providing a new proof to address a subtle gap in Khovanov's original argument, thereby clarifying the algebraic structure underlying link invariants.

Contribution

The authors present a new, rigorous proof that twists of Frobenius algebras produce isomorphic chain complexes in link homology, correcting a gap in Khovanov's 2006 proof.

Findings

Confirmed the isomorphism of chain complexes under Frobenius algebra twists

Provided a detailed analysis of circle configurations in link states

Strengthened the theoretical foundation of link homology invariants

Abstract

We discuss twists on Frobenius algebras in the context of link homology. In his paper in 2006, Khovanov asserted that a twist of a Frobenius algebra yields an isomorphic chain complex on each link diagram. Although the result has been widely accepted for nearly two decades, a subtle gap in the original proof was found in the induction step of the construction of the isomorphism. Following discussion with Khovanov, we decided to provide a new proof. Our proof is based on a detailed analysis of configurations of circles in each state.