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A Kontsevich integral of order $1$

Arnaud Mortier

Algebraic & Geometric Topology 22 (2022) 559–599

We define a 1–cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0–cocycle. It involves a 2–form that generalizes the Knizhnik–Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the algebra of chord diagrams and 1T–4T relations) is preserved between our integral and Vassiliev 1–cocycles, via a change of variable similar to the one that led Birman–Lin to discover the 4T relations. We explain how this construction is related to Cirio and Faria Martins’ categorification of the Knizhnik–Zamolodchikov connection.

To Joan S Birman and Xiao-Song Lin

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space of knots, cohomology, Kontsevich, Vassiliev, Teiblum–Turchin, Knizhnik–Zamolodchikov, chord diagrams, 4T relation
Mathematical Subject Classification 2010
Primary: 55T25, 57M25
Secondary: 53C29
Received: 18 May 2019
Revised: 21 December 2020
Accepted: 30 January 2021
Published: 3 August 2022
Arnaud Mortier
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen Normandie