Goldman–Turaev formality from the Kontsevich integral

Abstract

We present a new solution to the formality problem for the framed Goldman–Turaev Lie bialgebra in genus zero, constructing Goldman–Turaev homomorphic expansions (formality isomorphisms) from the Kontsevich integral. Our proof uses a three-dimensional derivation of the Goldman–Turaev Lie bialgebra arising from a low-degree Vassiliev quotient – the emergent quotient – of tangles in a thickened punctured disk modulo a Conway skein relation. This is in contrast to Massuyeau’s 2018 proof using braids. A feature of our approach is a general conceptual framework which is applied to prove the compatibility of the homomorphic expansion with both the Goldman bracket and the technically challenging Turaev cobracket.