We define a
–cocycle
in the space of long knots that is a natural generalization of the Kontsevich integral seen as a
–cocycle. It
involves a
–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
T–T
relations) is preserved between our integral and Vassiliev
–cocycles,
via a change of variable similar to the one that led Birman–Lin to discover the
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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