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arXiv · 2509.20983

Goldman-Turaev formality from the Kontsevich integral

Dror Bar-Natan·Zsuzsanna Dancso·Tamara Hogan·Jessica Liu·Nancy Scherich
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Abstract

We present a new solution to the formality problem for the framed Goldman--Turaev Lie bialgebra, constructing Goldman-Turaev homomorphic expansions (formality isomorphisms) from the Kontsevich integral. Our proof uses a three dimensional derivation of the Goldman-Turaev Lie biaglebra arising from a low-degree Vassiliev quotient -- the {\em 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.

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Dror Bar-Natan, Zsuzsanna Dancso, Tamara Hogan, Jessica Liu, Nancy Scherich. 2025-09-25. Goldman-Turaev formality from the Kontsevich integral. https://arxiv.org/abs/2509.20983

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