1.3 Integrability conditions, $\ker\delta$, lassoing singular points, and four-term relations

Next, we wish to find conditions that a ``potential top derivative'' has to satisfy in order to actually be a top derivative. More precisely, we wish to find conditions that a functional $W\in\bar{\mathcal K}^\star_n$ has to satisfy in order to be $\partial^nV$ for some invariant V. A first condition is that W must be ``integrable once''; namely, there has to be some $W^1\in{\mathcal K}^\star_{n-1}$ with $W=\partial W^1$. Another quick glance at (2), and we see that W is integrable once iff it vanishes on $\ker\delta$, which is the same as requiring that Wdescends to ${\mathcal A}_n = {\mathcal A}_n({\mathcal K}) := \bar{\mathcal K}_n/\pi(\ker\delta) = {\mathcal K}_n/(\operatorname{im}\delta+\ker\delta)$ (there should be no confusion regarding the identities of the $\delta$'s involved). Often elements of ${\mathcal A}^\star_n$ are refered to as ``weight systems''. A more accurate name would be ``once-integrable weight systems''.

We see that it is necessary to understand $\ker\delta$. In Figure 4 we show a family of members of $\ker\delta$, the ``Topological 4-Term'' (T4T) relations. Figure 5 explains how they arise from ``lassoing a singular point''. The following theorem says that this is all:

  

Figure 4: A Topological 4-Term (T4T) relation. Each of the four graphics in the picture represents a part of an n-singular knot (so there are n-2 additional singular points not shown), and, as usual in knot theory, the 4 singular knots in the equation are the same outside the region shown.

  

Figure 5: Lassoing a singular point: Each of the graphics represents an (n-1)-singular knot, but only one of the singularities is explicitly displayed. Start from the left-most graphic, pull the ``lasso'' under the displayed singular point, ``lasso'' the singular point by crossing each of the four arcs emenating from it one at a time, and pull the lasso back out, returning to the initial position. Each time an arc is crossed, the difference between ``before'' and ``after'' is the co-derivative of an n-singular knot (up to signs). The four n-singular knot thus obtained are the ones making the Topological 4-Term relation, and the co-derivative of their signed sum is the difference between the first and the last (n-1)-singular knot shown in this figure; namely, it is 0.

Theorem 1 (Stanford [St1])   The T4T relations of Figure 4 span $\ker\delta$. $\Box$

Pushing the T4T relations down to the level of symbols, we get the well-known 4T relations, which span $\pi(\ker\delta)$: (see e.g. [B-N1])

$$4T:\qquad \if ny \smash{\makebox[0pt]{\hspace{-0.5in} \rai... ...ox{-8pt}{ \input draws/4T.tex } \hspace{-1.9mm} \end{array}.$$

We thus find that ${\mathcal A}_n=(\text{chord diagrams})/(4T\text{\ relations})$, as usual in the theory of finite type invariants of knots.