Correction to: Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem

In Section 4 of [1] we introduce and study w-tangled foams.These are defined combinatorially, as a finitely generated circuit algebra with certain Reidemeister relations.For motivation, we present a local topological interpretation of w-tangled foams as tangled tubes in R 4 .Unfortunately, some errors have occurred in this interpretation, stemming from our lack of care around "2D orientations", as below.We wish to thank Yusuke Kuno and Haruko Miyazawa for noting these issues.• If foams are to have 2D orientations, they cannot be glued to each other unless these orientations match.This breaks the circuit algebra structure.This can be corrected by switching to coloured circuit algebras, in which strands and strand-ends are coloured by their 2D orientations, and gluings are allowed only if colours match.However, we choose a different resolution explained below.