The Kontsevich Integral for Knotted Trivalent Graphs

Paperlets Navigation Panel   Algebraic Knot Theory Algebraic Knot Theory - A Call for Action. The Kontsevich Integral for Knotted Trivalent Graphs. Knotted Trivalent Graphs are Finitely Presented. VS, TS and TG Algebras. The Alexander Polynomial of a Knotted Trivalent Graph. Some Questions About Trinions. Finite Type Invariants The Envelope of a Finite Type Invariant. The Envelope of The Alexander Polynomial. Some Quotients of A. Associators with Frozen Feet. The HOMFLY Braidor Algebra. Coloured Chord Diagrams. Khovanov Homology Mutation Invariance of Khovanov Homology. Khovanov Homology of Alternating Tangles. Other The Existence of the Exponential Function. The Many Faces of Alexander.

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

$Z_{0}(K)=\ \ \ \ \ \ \ \ \ \ \int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum _{m;\ t_{1}<\ldots <t_{m};\ P=\{(z_{i},z'_{i})\}}{\frac {(-1)^{\#P_{\downarrow }}}{(2\pi i)^{m}}}D_{P}\bigwedge _{i=1}^{m}{\frac {dz_{i}-dz'_{i}}{z_{i}-z'_{i}}}$

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

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 $\mu$.

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 $Z_{2}$.