A Kontsevich integral of order 1

Mortier, Arnaud

doi:10.2140/agt.2022.22.559

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A Kontsevich integral of order $1$

Arnaud Mortier

Algebraic & Geometric Topology 22 (2022) 559–599

DOI: 10.2140/agt.2022.22.559

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

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Keywords

space of knots, cohomology, Kontsevich, Vassiliev, Teiblum–Turchin, Knizhnik–Zamolodchikov, chord diagrams, 4T relation

Mathematical Subject Classification 2010

Primary: 55T25, 57M25

Secondary: 53C29

References

Publication

Received: 18 May 2019

Revised: 21 December 2020

Accepted: 30 January 2021

Published: 3 August 2022

Authors

Arnaud Mortier
Laboratoire de Mathématiques Nicolas Oresme Université de Caen Normandie Caen France

Volume 22, issue 2 (2022)