Summary. As an example where the scheme of Section 1.2 has been successfully used, and also in order to display some formulas for later use in this article, we give a very quick reminder of parenthesized tangles and the pentagon and hexagon relations and their syzygies, along the lines of [BN3,BN5,LM].
The papers [BN3,BN5,LM] introduce an algebraic context within which the scheme of Section 1.2 is used to construct a universal finite type invariants of links. The ``algebraic context'' there is the structure of a category with certain additional operations. Rather than defining everything in full, we will just recall some key notions, pictures and formulas here.
The category PaT of ``parenthesized tangles'', (the algebraic structure which we wish to represent, like in Section 1.2) is the category whose objects are parenthesizations such as or , and whose morphisms are tangles with parenthesized top and bottom. See the picture on the right, which also illustrates how parenthesized tangles are composed.
The category PaT carries some additional operations. The most interesting are the ``strand addition on the left/right'' operations, and the strand doubling operations (illustrated on the right). More details are in [BN3,BN5,LM].
Likewise, one can set up a category Pa of ``parenthesized chord diagrams'', that captures the ``symbols'' of ``singular'' parenthesized tangles as in Equation (1). The category Pa supports the same additional operations as PaT, and one may wish to look for structure preserving functors PaTPa which are ``essential'' in a sense similar to that of Equation (2). In [BN3], this is done following the same generators-relations-syzygies sequence as in Section 1.2:
Now let us assume that we already found and so that the
relations between them corresponding to and
satisfied up to degree 16 (say), and let and be the degree
17 errors in these equations (compare with Equation (4)).
That is, modulo degrees 18 and up we have (notaion as
in [BN3], compare with [BN3, Equations (10)
Proceeding as in Equation (5) we set
and with and of
degree 17, and like in Equation (6) we get (compare
with [BN3, Equations (12) and (13)]):
Thus we are interested in knowing whether the triple is in the image of the linear map