The Kontsevich Integral for Knotted Trivalent Graphs

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In Preparation

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



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

07-1352 Kontsevich Integral.png
Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<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}

We define the "naive Kontsevich integral" Z_1 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

07-1352 Renormalizing the Kontsevich Integral.png

After choosing a scale \mu and an infinitesimal \epsilon and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" Z_2 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 \mu, as follows from:

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

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

The Bad

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

Step 3 - The Corrected Kontsevich Integral

07-1352 Correctors Corrections Syzygies.png

By correcting caps and cups with factors n and u, and the two types of trivalent vertices with factors y and \lambda, we define the "corrected Kontsevich integral" Z=Z_3 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 \Phi_{KZ} is reproduced already by Z_2:

07-1352 Normalizing PhiKZ.png