DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Jones–Vassiliev coefficient vectors

Definition

Assume full AC for the library Jones invariant. An oriented link is a finite disjoint union of oriented smooth circles embedded in $S^{3}$ , considered up to ambient isotopy; an oriented classical knot is the one-component case. For a nonempty oriented link $L \subset S^{3}$ , let $V_{L} (x) \in Z [x^{\pm 1}]$ be the library-normalized Jones polynomial, with $V_{U} (x) = 1$ for the unknot $U$ ; write $V_{K}$ for its value on a knot $K$ . Here $x = t^{1 / 2}$ in the library convention $t = A^{- 4}$ , so an oriented skein triple satisfies $x^{- 2} V_{L_{+}} (x) - x^{2} V_{L_{-}} (x) = (x - x^{- 1}) V_{L_{0}} (x) .$ Set $p = x - x^{- 1}$ . By the unique Laurent decomposition, write $V_{K} (x) = \sum_{q \geq 0} (a_{q} (K) + b_{q} (K) x) p^{q},$ where $a_{q} (K), b_{q} (K) \in Z$ and only finitely many are nonzero. The Jones–Vassiliev coefficient vector of order $q$ is $f_{q} (K) : = (a_{q} (K), b_{q} (K))^{T},$ and $c_{q} (K) : = a_{q} (K) + b_{q} (K)$ . This is the algebraic coefficient-vector definition used by Carmi and Cohen. The claim that the coordinates of $f_{q}$ are finite-type invariants of order at most $q$ is a separate theorem and is not part of this definition.

Definition references

Unique Laurent decomposition

Library Jones normalization

Bracket/Jones comparison

Unframed Vassiliev finite-difference convention

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, §1.3 and Lemma 3.1

Kauffman, Combinatorial Knot Theory and the Jones Polynomial, §2

Roberts, Knots Knotes, §§4.3–4.4

Bar-Natan and Stoimenow, The Fundamental Theorem of Vassiliev Invariants, §1.1

Depends on

Used by

The unknot defeats a universal coefficient-bound extension Counterexample
A trefoil Jones–Vassiliev coefficient calculation Example
Jones polynomial unknot-detection status Remark

Dependency tree · two levels

34 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

Avishy Carmi and Eliahu Cohen, On trivial Jones–Vassiliev polynomials, arXiv:2606.22410v1, §1.3 and Lemma 3.1 (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)
Justin Roberts, Knots Knotes, §§4.3–4.4 (standard reference, not scraped)
Dror Bar-Natan and Alexander Stoimenow, The Fundamental Theorem of Vassiliev Invariants, arXiv:q-alg/9702009v1, §1.1 (standard reference, not scraped)