#### Volume 22, issue 2 (2022)

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A Kontsevich integral of order $1$

### Arnaud Mortier

Algebraic & Geometric Topology 22 (2022) 559–599
##### Abstract

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 $1$T–$4$T 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 $4$T 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
##### Keywords
space of knots, cohomology, Kontsevich, Vassiliev, Teiblum–Turchin, Knizhnik–Zamolodchikov, chord diagrams, 4T relation
##### Mathematical Subject Classification 2010
Primary: 55T25, 57M25
Secondary: 53C29