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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Open questions in knot polynomials and surgery

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Unique Laurent decomposition for the Jones–Vassiliev basis

Statement

Let $x$ be an invertible indeterminate and put $p = x - x^{- 1}$ . Every Laurent polynomial $P (x) \in Z [x, x^{- 1}]$ has a unique expression $P (x) = a (p) + b (p) x$ with $a, b \in Z [p]$ , where $Z [p]$ is the ring of finite integer-coefficient polynomials in the indeterminate $p$ .

Statement references

Polynomial rings over commutative rings

Quotient rings

Division by a monic polynomial

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, Lemma 3.1 and Appendix B.1

Donaldson, Math 120B Notes, Section 22

Facts & Assumptions

Given: The commutative polynomial ring $R = Z [p]$ , the monic polynomial $G (X) = X^{2} - p X - 1 \in R [X]$ , and the Laurent polynomial ring $Z [x, x^{- 1}]$ .

[F1]

The polynomial ring $R [X]$ consists of finite-support coefficient sequences with coefficientwise addition and finite convolution multiplication (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

[F2]

If $R$ is commutative, then $R [X]$ with these operations is a commutative ring (Polynomial convolution makes $R [x]$ a commutative ring containing $R$ as its constant subring).

[F3]

The ideal generated by $G$ is a two-sided ideal; the quotient ring consists of its additive cosets with multiplication induced from $R [X]$ (The ideal generated by a subset and principal ideals, The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ).

[F4]

Division by the monic degree-two polynomial $G$ gives each polynomial in $R [X]$ a unique remainder of degree less than two (Division by a monic polynomial over a commutative ring).

Proof

1.1givenF1F2F3F4construct

Let $Q = R [X] / (G)$ . By [F1] and [F2], $R [X]$ is a commutative polynomial ring, and [F3] makes $(G)$ a two-sided ideal and $Q$ a quotient ring. Since $G$ is monic of degree two, [F4] gives every class in $Q$ a representative $a (p) + b (p) X$ of degree at most one in $X$ . If two such representatives define the same class, their difference is a multiple of $G$ and has degree less than two; uniqueness in [F4] forces the difference to be zero. Thus this representative is unique.

2.1givenF1F2F3step 1.1algebra∎

Define $ϕ : R [X] \to Z [x, x^{- 1}]$ on finite polynomials by $ϕ (\sum_{j} c_{j} (p) X^{j}) = \sum_{j} c_{j} (x - x^{- 1}) x^{j}$ . Substitution and finite convolution make $ϕ$ a ring homomorphism, and $ϕ (G) = x^{2} - (x - x^{- 1}) x - 1 = 0$ , so $ϕ$ is constant on cosets of $(G)$ and induces $\overset{ˉ}{ϕ} : Q \to Z [x, x^{- 1}]$ . In $Q$ , $X (X - p) = 1$ , so $X$ is a unit with inverse $X - p$ . Define $ψ : Z [x, x^{- 1}] \to Q$ by $ψ (\sum_{k} n_{k} x^{k}) = \sum_{k \geq 0} n_{k} X^{k} + \sum_{k < 0} n_{k} (X - p)^{- k}$ ; because $X - p = X^{- 1}$ , finite Laurent multiplication is preserved. The composite $\overset{ˉ}{ϕ} ψ$ fixes $x$ , while $ψ \overset{ˉ}{ϕ}$ fixes $X$ and $p$ ; since $Q$ is generated by $X$ and $p$ , the maps are inverse. The unique representatives from step 1.1 therefore give the asserted unique Laurent decomposition.

DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

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

RemarkRemark: Literature-sourcedProof: Not applicableverified 2026-10-22 (gpt-6.1-sol)Open item page →

Jones polynomial unknot-detection status

Historical question

Assume full AC for the library Jones invariant and ${A C}_{ω}$ for the crossing number. A classical knot is a smooth embedding $S^{1} ↪ S^{3}$ considered up to ambient isotopy; $U$ is the unknot represented by a crossing-free circle, and “nontrivial” means it is not ambient isotopic to $U$ . A regular diagram is a generic plane projection with only transverse double points, each marked with over/under information. Let $c (K)$ be the least crossing count among regular diagrams representing $K$ . Let $V_{K} (x)$ be the library-normalized Jones polynomial with $V_{U} (x) = 1$ , where $x = t^{1 / 2}$ and $t = A^{- 4}$ . Set $p = x - x^{- 1}$ and write the unique expansion $V_{K} (x) = \sum_{q \geq 0} (a_{q} (K) + b_{q} (K) x) p^{q},$ where the integer coefficients have finite support; set $f_{q} (K) = (a_{q} (K), b_{q} (K))^{T}$ . The historical Jones unknot-detection question is whether $V_{K} (x) = 1$ implies $K = U$ .

Historical question references

Jones normalization and coefficient notation

Crossing number

Roberts, Knots Knotes, §4.4.8

Remark

Roberts's 1999 notes record the implication as a conjecture. Carmi and Cohen's arXiv:2606.22410v1, submitted on 2026-06-21, claims that every nontrivial classical knot has a nonzero coefficient vector $f_{q} (K)$ for some integer $q$ with $2 \leq q \leq 3 c (K)$ , and claims $V_{K} (x) = 1 \Rightarrow K = U$ . This scaffold has read the stated result and the paper's proof route but has not independently validated the claimed proof; it asserts neither proof acceptance nor a definitive current open status.

The source's machine argument still requires an independent diagram-level audit of Proposition 5.12, particularly its exceptional-state collapse, smoothing proxy, and two-machine locality. Its numerical estimate $B_{a u x} \leq B_{h o s t} + 12$ is consistent with the stated overhead $κ_{m a c h} = 11$ : the argument uses the next layer $r + 1$ after $r > B_{h o s t} + 11$ . This arithmetic check does not verify the geometric reductions. No Jones-detection theorem is registered by this scaffold.

Remark references

Carmi and Cohen, On trivial Jones–Vassiliev polynomials, §§1.3, 3, 5.6, 6.2–6.3