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Vassiliev invariants induce weight systems

Statement

Let $R$ be a commutative ring and $n \geq 0$ . For an unframed $R$ -valued invariant $V$ of type at most $n$ , the degree- $n$ derivative induces an $R$ -linear functional on degree- $n$ chord diagrams modulo 4T and 1T. For a framed $R$ -valued invariant $V$ of type at most $n$ , assume ${A C}_{ω}$ ; its degree- $n$ derivative induces an $R$ -linear functional on degree- $n$ chord diagrams modulo 4T. Framed derivatives are not subject to a universal 1T vanishing law: integer-valued framed self-linking has value $2$ on the one-chord diagram with blackboard framing.

Statement references

Bar-Natan and Stoimenow, §1.1, pp. 3–5

Chmutov, Duzhin, and Mostovoy, §§3.4 and 4.1–4.2, pp. 70–73 and 89–94

Facts & Assumptions

Given: A commutative ring $R$ , an integer $n \geq 0$ , and an $R$ -valued invariant $V$ of type at most $n$ . The 4T and unframed 1T conclusions are choice-free. The framed case and its non-1T witness assume ${A C}_{ω}$ as in [A1].

[F1]

Singular knots and Vassiliev derivatives: the $n$ -fold derivative is the alternating sum over independent resolutions; framed type bounds also count framing zeroes as singularities.

[F2]

Framed chord diagrams and four-term relations: both framed and unframed symbol modules are generated by ordinary degree- $n$ chord diagrams; 4T is imposed in both and 1T only in the unframed module.

[F3]

The four-term relation from singular Reidemeister III: in both categories the derivative value is representative-independent and annihilates every 4T generator.

[F4]

The one-term relation and framing: unframed derivatives vanish on diagrams with an isolated chord; under ${A C}_{ω}$ , framed self-linking has type at most $1$ and value $2$ on the one-chord blackboard-framed diagram.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ selects one element from each set in a sequence of nonempty sets; it is assumed for the framed normal-bundle branch and the framed witness inherited from [F4], not for the unframed 4T or 1T arguments.

Proof

Given: As above, with ${A C}_{ω}$ added when working in the framed category and when invoking the framed witness clause of [F4].

1.1givenA1F1F3chooseconstruct

For each degree- $n$ diagram $D$ , choose an $n$ -singular representative $K_{D}$ and set $w_{V} (D) = D_{n} V (K_{D})$ . By [F3] the value is independent of the representative and, in the framed case, of its zero-free framing class. Thus this defines a function on degree- $n$ diagrams. Extending on the free $R$ -module by $R$ -linearity defines a linear functional.

2.1A1F2F3step 1.1algebra

By [F3], $w_{V} (D_{1}) - w_{V} (D_{2}) + w_{V} (D_{3}) - w_{V} (D_{4}) = 0$ for every 4T generator. Thus the functional annihilates the 4T submodule and factors through the framed quotient by the definition in [F2].

3.1F2F4A1step 1.1algebra∎

If $V$ is unframed, [F4] makes the functional vanish on every 1T generator as well, so it factors through the unframed quotient in [F2]. In the framed branch under ${A C}_{ω}$ , [F4] and [A1] give a type-at-most-one framed self-linking example with one-chord value $2$ ; since $2 \neq 0$ in $Z$ , no universal framed 1T relation can be added.

Depends on

Used by

The framed degree-one weight system does not pass through 1T Counterexample
Universality of the Kontsevich integral Theorem

Dependency tree · two levels

19 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

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