The Kontsevich Integral for Knotted Trivalent Graphs

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In Preparation

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

Contents

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

We define the "naive Kontsevich integral" Z_1 of a knotted trivalent graph or a slice thereof.

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 \mu 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.

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.

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 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.

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

The Drinfel'd KZ associator is reproduced already by Z_2.