We want extend our definition of Kontsevich integral to tangles, though it preserves the tangle composition, it does not preserve the horizontal stacking, because swapping the endpoints will affect the associated chord diagram.
To solve this, we will make
-Parametrised Tangle An
-parametrised tangle is a unit wide tangle with parenthesised words in as ends, and any adjacent endpoints within closed parentheses are apart.
e.g.
Parenthesised Oriented Tangles
A parenthesised oriented tangle category
is a category with
- objects being parenthesised words in
, - morphisms being
-parametrised tangles with objects as ends
e.g.
But it turns out this is not finitely generated, so we need to add one more structure, strand doubling.
Theorem
There is a one to one correspondence between the set of formality maps for
and the set of Drinfeld associators.