Correction to the article: Floer homology and splicing knot complements

TL;DR

This paper corrects previous errors in splicing formulas related to knot Floer homology, specifically addressing the role of an involution that was previously assumed trivial, thereby refining the mathematical understanding of knot complements.

Contribution

It identifies and corrects mistakes in earlier splicing formulas by properly accounting for the involution on knot Floer homology, improving the accuracy of the theory.

Findings

Corrected splicing formulas incorporating the involution.

Clarified the role of the involution in knot Floer homology.

Enhanced the mathematical framework for knot complement analysis.

Abstract

This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by one full twist around $K$ is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places.