1.3 Parenthesized tangles

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 $B_4$ in Section 1.2) is the category whose objects are parenthesizations such as $(((\cdot\cdot)\cdot)\cdot)$ or $((\cdot\cdot)(\cdot\cdot))$, 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${\mathcal A}$ of ``parenthesized chord diagrams'', that captures the ``symbols'' of ``singular'' parenthesized tangles as in Equation (1). The category Pa${\mathcal A}$ supports the same additional operations as PaT, and one may wish to look for structure preserving functors $Z:$PaT$\to$Pa${\mathcal A}$ 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:

1.3.1 Generators

The category PaT is generated by the morphisms $\Associator$ and $\slashoverback$. We set $\tilde{\Phi}=Z(\Associator)$ and $\tilde{R}=Z(\slashoverback)$ and then reconsider these morphisms in Pa${\mathcal A}$ as elements $\Phi\in{\mathcal A}(\uparrow_3)$ and $R\in{\mathcal A}(\uparrow_2)$, where ${\mathcal A}(\uparrow_n)$ denotes the usual space of chord diagrams modulo $AS$, $IHX$ and $STU$ relations on a skeleton made of $n$ vertical lines.

1.3.2 Relations

There are several relations beween $\Associator$ and $\slashoverback$ as elements of the algebraic structure Pa${\mathcal A}$, as listed in [BN3]. The most prominent of those are the pentagon $\pentagon$ and the two hexagon relations $\hexagon _\pm$, displayed in Figures 2 and 3.

Figure 2: The pentagon relation $\pentagon$ and its tensor-category-theoretical origin.

Figure 3: The positive and negative hexagon relations $\hexagon _\pm$ and their tensor-categorical origin.

Now let us assume that we already found $R$ and $\Phi$ so that the relations between them corresponding to $\pentagon$ and $\hexagon _\pm$ are satisfied up to degree 16 (say), and let $\mu$ and $\psi_\pm$ 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) and (11)]):

$$\mu = \Phi^{123}\cdot(1\otimes\Delta 1)(\Phi)\cdot\Phi^{234} -(\Delta\otimes 1\otimes 1)(\Phi)\cdot(1\otimes 1\otimes\Delta)(\Phi),$$

$$\psi_\pm = \Phi^{123}\cdot(R^{\pm 1})^{23}(\Phi^{-1})^{132}\cdot(R^{\pm 1})^{13} \cdot\Phi^{312} - (\Delta\otimes 1)(R^{\mp 1})$$

Proceeding as in Equation (5) we set $\Phi'=\Phi+\varphi$ and $R'=R+r$ with $\varphi$ and $r$ of degree 17, and like in Equation (6) we get (compare with [BN3, Equations (12) and (13)]):

$$\mu' = \mu + \varphi^{234}-(\Delta\otimes 1\otimes 1)(\varphi) +(1\otimes\Delta 1)(\varphi)-(1\otimes 1\otimes\Delta)(\varphi) +\varphi^{123}$$

$$\psi'_\pm = \psi_\pm + \varphi^{123}-\varphi^{132}+\varphi^{312}\pm\left( r^{23}-(\Delta\otimes 1)(r)+r^{13} \right).$$

Thus we are interested in knowing whether the triple $E:=(\mu,\psi_\pm)$ is in the image of the linear map

$$d^r:\left(\begin{array}{c} \varphi r \end{array}\right) \maps... ...^{312}\pm\left( r^{23}-(\Delta\otimes 1)(r)+r^{13} \right) \end{array}\right).$$

1.3.3 Syzygies

On like in Section 1.2, the trick is to use syzygies between the $\pentagon$ and $\hexagon_{\pm}$ relations to reduce the problem to the computation of a homology group $H:=\ker d^s/\operatorname{im}d^r$ where $d^s$ is some other linear map, for which $d^s\circ d^r=0$ and $d^sE=0$. Again, this was carried out in full in [BN3]. Here we only reproduce the four syzygies we need to use (see Figure 4) and the resulting map $d^s$ (the four components of $d^s$ correspond to the syzygies in Figure 4 in the order $\tiny\begin{array}{\vert c\vert c\vert}\hline 1&2 \hline 3&4 \hline\end{array}$; the symbol $(\Delta 111)$ denotes $(\Delta\otimes 1\otimes 1\otimes 1)$ etc.; compare with [BN3, Equations (15-19)]):

$$d^s:\left(\begin{array}{c} \mu \psi_\pm \end{array}\right) \m... ...} \\ \psi_+^{213}-\psi_+^{123}+\psi_-^{231}-\psi_-^{321} \end{array}\right).$$

Figure: Four syzygies in PaT. The vertices in these pictures correspond to objects of PaT, the edges to morphisms, the faces to relations and hence each of the four polyhedra is a single syzygy. Further details are in [BN3,BN5].