Problem 3.19 — Find a topological construction of the polynomial P ′ K given above.

Definitions and mathematical statement

Problem 3.19 asks for a topological construction of a knot polynomial $P'_K$ "given above." The defining formula and normalization of $P'_K$ are absent from this record. Consequently this extract does not identify which polynomial must be constructed, and it cannot justify substituting another knot polynomial or a chosen loop coefficient.

The subsequent supplied context concerns the general loop expansion. Let $R$ be a field, for example ${\mathbb{Q}}$, and $S\subset R(t)$ a subring invariant under $t\mapsto t^{-1}$. A labeled Jacobi diagram on $\varnothing$ is a vertex-oriented trivalent graph, with each edge carrying a local orientation and a label in $S$. The excerpt defines $\mathcal A_S(\varnothing;R)$ as the $R$-vector space spanned by these diagrams modulo AS, IHX, multilinear and push relations. The multilinear and push relations are supplied only as missing Figures 13 and 14; their complete diagrammatic definitions cannot be reconstructed from the surviving text. The excerpt calls half the number of trivalent vertices the loop-degree. This is retained as its convention, without identifying it with the first Betti number of a connected graph.

For a polynomial $A(t)$ with $A(1)=1$ and $A(t)=A(t^{-1})$, it describes a diagrammatic map \[ \mathcal A_{{\mathbb{Q}}[t^{\pm1},1/A(t)]}(\varnothing;{\mathbb{Q}}) \longrightarrow\mathcal B.\tag{29} \] The surviving coefficient prescription is \[ f(e^h)=\sum_{k\ge0}c_kh^k \quad\left(f(t)\in{\mathbb{Q}}[t^{\pm1},1/A(t)]\right). \] The images of the labeled edges are missing pictures. In the special case it writes \[ \mathcal A_{{\mathbb{Q}}[t^{\pm1}]}(\varnothing;{\mathbb{Q}}) \longrightarrow\mathcal B,\tag{30} \] with the displayed diagram series having coefficients $1,1,1/2,\ldots,1/n!,\ldots$; again the diagrams themselves are absent.

The context states that a rational invariant \[ Z^{\rm rat}(K)\in \mathcal A_{{\mathbb{Q}}[t^{\pm1},1/\Delta_K(t)]}(\varnothing;{\mathbb{Q}}) \] is carried by (29) to $\log_{\sqcup}Z(K)$, where $Z$ is the Kontsevich invariant and $\Delta_K$ is the knot's Alexander polynomial in the source normalization. In particular, when $\Delta_K(t)=1$, it places $Z^{\rm rat}(K)$ in $\mathcal A_{{\mathbb{Q}}[t^{\pm1}]}(\varnothing;{\mathbb{Q}})$. The text attributes existence to [231] and canonicality to [139]. These contextual assertions do not supply the missing definition of $P'_K$; target completion and all normalizations also require source review.

Short English statement

Construct the particular polynomial $P'_K$ directly from topology. Its preceding definition is missing, so the requested invariant cannot be identified precisely. The following loop-expansion discussion is retained with its coefficient rings, rational Kontsevich invariant, and missing diagrammatic relations explicitly recorded.

Sources

• Local UnsolvedMath registry, permanent ID $10400069$, exact source code catalogue entry, Problem 3.19. No individual proposer or proposed year is supplied.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 3.19 and following loop-expansion context, PDF page 69: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. The monograph's bold headings distinguish 231 numbered items from surrounding references.
• Internal source references [231] for existence of $Z^{\rm rat}$ and [139] for canonicality; bibliography entries are not supplied. Figures 13 and 14 and edge-expansion pictures in (29)--(30) do not survive in this record.
• Extraction pdf-font-structure, accessed 2026-07-29. Source status NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied research classification OPEN-TRIAGE calls the topological-construction question not obviously resolved.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.