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Finite-type truncations of the Kontsevich integral

Statement

Assume countable choice, and let $I_{m} (K)$ be the degree- $m$ homogeneous coefficient of the normalized unframed Kontsevich series $I (K)$ of a smooth oriented knot type $K$ . For $n \geq 0$ , put $I_{\leq n} (K) = \sum_{m = 0}^{n} I_{m} (K)$ . A function on knot types is of type at most $n$ when its alternating resolution sum on every $(n + 1)$ -singular knot is zero. Then $I_{\leq n}$ is a type-at-most- $n$ invariant with values in the complexified direct sum $⨁_{m = 0}^{n} (A_{m}^{u n} \otimes_{Q} C)$ of the rational unframed degree- $m$ chord-diagram modules. For every $n$ -singular knot with chord diagram $D$ , its $n$ -fold finite difference satisfies $D_{n} I_{\leq n} (K) = D$ . In particular, the degree- $n$ symbol is the identity on rational chord diagrams after extending scalars to $C$ ; no rational-coefficient claim is made about $I_{m} (K)$ .

Statement references

Finite differences and type at most n

Leading symbol of the normalized integral

Ambient-isotopy invariance under countable choice

Chmutov, Duzhin, and Mostovoy, §8.8.1, printed pp. 239–240

Bar-Natan and Stoimenow, §2.6, PDF pp. 16–17

Facts & Assumptions

Given: Countable choice, a degree cutoff $n$ , and the unframed normalized series of a strict Morse presentation of each oriented knot type.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice is assumed only through the ambient-isotopy invariance supplier.

[F1]

The Kontsevich integral on parenthesized braids: $I (K)$ is the degreewise series in the completed unframed 4T+1T chord-diagram module.

[F2]

Singular knots and Vassiliev derivatives: $D_{k} F (K)$ is the alternating sum over the $2^{k}$ resolutions of a $k$ -singular knot; type at most $n$ means $D_{n + 1} F = 0$ on every $(n + 1)$ -singular knot.

[F3]

The Kontsevich integral induces the identity on the associated graded: on a $k$ -singular knot with chord diagram $D$ , $D_{k} I (K)$ has zero degrees below $k$ and degree- $k$ component exactly $D$ .

[F4]

Braid and knot isotopy invariance of the normalized integral: under countable choice, $I (K)$ is independent of the strict Morse presentation and is invariant under ambient isotopy of oriented knots.

[F5]

Framed chord diagrams and four-term relations: each degreewise unframed chord-diagram module is the rational 4T+1T quotient, and its complexification is the target of $I_{m}$ .

Proof

Given: As above. Let $π_{\leq n}$ denote projection to homogeneous degrees $0, 1, \dots, n$ ; it is linear and commutes with finite sums.

1.1A1F1F4F5algebra

By [F4] the normalized series is invariant under ambient isotopy and independent of its strict Morse presentation. Applying the degree projection $π_{\leq n}$ preserves equality, so $I_{\leq n} = π_{\leq n} I$ is a well-defined invariant on oriented knot types with the stated target.

2.1F2F3step 1.1algebra

Let $K$ be an $(n + 1)$ -singular knot with chord diagram $D$ . By [F3], every homogeneous component of $D_{n + 1} I (K)$ in degree at most $n$ is zero. Since $π_{\leq n}$ commutes with the alternating resolution sum, $D_{n + 1} I_{\leq n} (K) = π_{\leq n} D_{n + 1} I (K) = 0$ . This is exactly the type-at-most- $n$ condition from [F2].

3.1F1F2F3step 1.1step 2.1algebra∎

Let $K$ instead be $n$ -singular with diagram $D$ . By [F3], the projection of $D_{n} I (K)$ to degrees at most $n$ has no terms below degree $n$ and has degree- $n$ term $D$ . Therefore $D_{n} I_{\leq n} (K) = π_{\leq n} D_{n} I (K) = D$ . For $n = 0$ , the same statement is the degree-zero unit term of [F1]. Thus the degree- $n$ symbol of the truncation is the identity on rational chord diagrams, with coefficients regarded in their complexification.

Depends on

Used by

Universality of the Kontsevich integral Theorem

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Introduction to Vassiliev Knot Invariants, §8.8.1 (standard reference, not scraped)
The Fundamental Theorem of Vassiliev Invariants, §2.6 (standard reference, not scraped)