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Universality of the Kontsevich integral

Statement

Assume countable choice. Let $V_{n}^{C}$ be the $C$ -vector space of ambient-isotopy invariants of oriented knots with values in $C$ and type at most $n$ , where type at most $n$ means that the alternating resolution sum on every $(n + 1)$ -singular knot is zero; set $V_{- 1}^{C} = 0$ . Let $A_{n}^{u n}$ be the rational vector space on degree- $n$ circle chord diagrams modulo 4T and 1T; a weight system is a $Q$ -linear functional on this space. For an $n$ -singular knot with chord diagram $D$ , define $σ_{n} ([V]) (D)$ to be the $n$ -fold alternating resolution sum of $V$ on that knot. Let $I_{\leq n} (K) = \sum_{j = 0}^{n} I_{j} (K)$ be the degree- $n$ truncation of the normalized unframed Kontsevich series. The degree- $n$ symbol map

σ_{n} : V_{n}^{C} / V_{n - 1}^{C} ⟶ {Hom}_{Q} (A_{n}^{u n}, C)

is surjective. In particular, every rational weight system has a complex-valued finite-type lift; no rationality claim is made about that lift or about the coefficients of $I (K)$ . Moreover, for every $n$ and every $V \in V_{n}^{C}$ there is a complex-linear functional $Λ_{V}$ on $⨁_{j = 0}^{n} (A_{j}^{u n} \otimes_{Q} C)$ whose degree- $n$ restriction is the degree- $n$ symbol of $V$ and such that

V (K) = Λ_{V} (I_{\leq n} (K))

for every oriented knot type $K$ . This recovery statement does not imply that distinct knot types have distinct Kontsevich series.

Statement references

Finite differences and type at most n

Unframed 4T+1T chord-diagram module

Weight systems induced by finite-type invariants

Finite-type truncations and identity symbols

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

Bar-Natan and Stoimenow, §§1.1 and 2.6, PDF pp. 3–5 and 16–17

Facts & Assumptions

Given: Countable choice, the unframed finite-difference convention, and the normalized Kontsevich series with coefficients in the complexified chord-diagram completion.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice is the only axiom needed here, through ambient-isotopy invariance of the normalized series.

[F1]

Singular knots and Vassiliev derivatives: an invariant is of type at most $n$ when its $(n + 1)$ -fold alternating resolution sum vanishes on every $(n + 1)$ -singular knot.

[F2]

Framed chord diagrams and four-term relations: $A_{j}^{u n}$ is the rational vector space on degree- $j$ circle chord diagrams modulo 4T and 1T.

[F3]

Vassiliev invariants induce weight systems: for an unframed invariant of type at most $n$ , the $n$ -fold derivative is a well-defined rational-linear weight system on $A_{n}^{u n}$ ; the conclusion holds for complex-valued invariants as well.

[F4]

Finite-type truncations of the Kontsevich integral: $I_{\leq n}$ is type at most $n$ , and on an $n$ -singular knot with diagram $D$ its $n$ -fold derivative is $D$ .

[F5]

The Kontsevich integral induces the identity on the associated graded: for each $k$ -singular knot, $D_{k} I$ has no homogeneous terms below degree $k$ and its degree- $k$ term is the chord diagram of that knot.

[F6]

Braid and knot isotopy invariance of the normalized integral: under countable choice, each coefficient $I_{j} (K)$ is a well-defined ambient-isotopy invariant of oriented knot types.

Proof

Given: As above. Extend each rational weight system complex-linearly to the complexified chord-diagram module.

1.1A1F2F4F5F6constructalgebra

Let $W \in {Hom}_{Q} (A_{n}^{u n}, C)$ be any weight system. Define $V_{W} (K) = W (I_{n} (K))$ , using the complex-linear extension of $W$ . By [F6] this is an ambient-isotopy invariant. Since $I_{\leq n}$ is of type at most $n$ by [F4], each homogeneous coefficient, and hence $V_{W}$ , is of type at most $n$ . For an $n$ -singular knot with diagram $D$ , [F5] gives $D_{n} V_{W} (K) = W (D)$ . Thus $σ_{n} ([V_{W}]) = W$ , proving surjectivity; in particular this holds for rational-valued weight systems.

2.1F1F3F4F5F6step 1.1constructalgebra∎

Let $V \in V_{n}^{C}$ . Put $R^{(n)} = V$ . For $k = n, n - 1, \dots, 0$ , define $W_{k} = σ_{k} ([R^{(k)}])$ using [F3], and set $R^{(k - 1)} (K) = R^{(k)} (K) - W_{k} (I_{k} (K))$ . The type bound in [F4] makes $W_{k} \circ I_{k}$ type at most $k$ . For a $k$ -singular knot with diagram $D$ , [F5] gives $D_{k} (W_{k} \circ I_{k}) (K) = W_{k} (D) = D_{k} R^{(k)} (K)$ . Therefore $R^{(k - 1)}$ has zero $k$ -fold derivative and is of type at most $k - 1$ . At $k = 0$ , the residual is type at most $- 1$ , hence is zero by the convention in [F1]. Consequently $V (K) = \sum_{k = 0}^{n} W_{k} (I_{k} (K))$ . Define $Λ_{V}$ on the degree- $k$ summand by $W_{k}$ and extend complex-linearly over the finite direct sum. This gives $V (K) = Λ_{V} (I_{\leq n} (K))$ for every knot type. No step uses separation of knot types by $I$ .

Depends on

Used by

Conway coefficients and the degree-two Kontsevich symbol Example

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Sources

Introduction to Vassiliev Knot Invariants, §8.8.1 (standard reference, not scraped)
The Fundamental Theorem of Vassiliev Invariants, §§1.1 and 2.6 (standard reference, not scraped)