The Kontsevich Integral for Knotted Trivalent Graphs

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Abstract

We construct a (very) well-behaved invariant of knotted trivalent graphs using only the Kontsevich integral, in three steps.

Step 1 - The Naive Kontsevich Integral

[math]\displaystyle{ Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1\lt \ldots\lt t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i} }[/math]

We define the "naive Kontsevich integral" [math]\displaystyle{ Z_1 }[/math] of a knotted trivalent graph or a slice thereof as in the "standard" picture above, except generalized to graphs in the obvious manner.

The Good

• It has a factorization property.
• For the "braid-like" factors, it has invariance under horizontal deformations.
• It is morally a universal finite type invariant.

The Bad

• It is divergent.

Step 2 - The Renormalized Kontsevich Integral

After choosing a scale [math]\displaystyle{ \mu }[/math] and an infinitesimal [math]\displaystyle{ \epsilon }[/math] and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" [math]\displaystyle{ Z_2 }[/math] of a knotted trivalent graph or a slice thereof as summarized by the picture above.

The Good

• It retains all the good properties of the naive Kontsevich integral.
• It is convergent.
• It is invariant under rigid motions of critical points and trivalent vertices.
• It has sensible behaviour under vertical connect sum, delete and unzip operations.
• It has a sensible behaviour under the changing of the scale [math]\displaystyle{ \mu }[/math], as follows from:

Claim. Let [math]\displaystyle{ w_1,\ldots,w_n }[/math] be distinct complex numbers and let [math]\displaystyle{ \beta }[/math] be another complex number. Let [math]\displaystyle{ B }[/math] be the ([math]\displaystyle{ n }[/math]-strand) braid "rescaling braid" which is the image of the map [math]\displaystyle{ [0,1]\to[0,1]\times{\mathbb C}^n }[/math] defined by [math]\displaystyle{ t\mapsto(t, e^{\beta t}w_1,\ldots,e^{\beta t}w_n) }[/math]. Finally, in [math]\displaystyle{ {\mathcal A}(\uparrow_n) }[/math] let [math]\displaystyle{ c }[/math] be the "sum of all horizontal chords"; [math]\displaystyle{ c=\sum_{1\leq i\lt j\leq n}t^{ij} }[/math]. Then up to normalization factors which we are getting right with probability [math]\displaystyle{ \frac{1}{2^k} }[/math] for some small [math]\displaystyle{ k\in{\mathbb N} }[/math],

[math]\displaystyle{ Z(B)=\exp\frac{\beta c}{2\pi i}\in{\mathcal A}(\uparrow_n) }[/math].

The Bad

• It is not invariant under non-rigid motions of vertices and critical points.

Step 3 - The Corrected Kontsevich Integral

By correcting caps and cups with factors [math]\displaystyle{ n }[/math] and [math]\displaystyle{ u }[/math], and the two types of trivalent vertices with factors [math]\displaystyle{ y }[/math] and [math]\displaystyle{ \lambda }[/math], we define the "corrected Kontsevich integral" [math]\displaystyle{ Z=Z_3 }[/math] of a knotted trivalent graph or a slice thereof. It has all the good properties we can wish for. The details still need to be worked out here!

Aside - The Relationship with Drinfel'd's KZ Associator

The Drinfel'd KZ associator [math]\displaystyle{ \Phi_{KZ} }[/math] is reproduced already by [math]\displaystyle{ Z_2 }[/math]: