A Kontsevich integral of order 1

Abstract

We define a 1-cocycle in the space of long knots that is a natural generalisation of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalises the Knizhnik-Zamolodchikov connection. Similarly to the Kontsevich integral, it lives in a space of chord diagrams of the same kind as those that make the principal parts of Vassiliev's 1-cocycles. Moreover, up to a change of variable similar to the one that led Birman-Lin to discover the 4T relations, we show that the relations defining our space, which allow the integral to be finite and invariant, are dual to the maps that define Vassiliev's cohomology in degree 1.