Conjecture 3.8 — (D.

Definitions and mathematical statement

Let $O$ be the trivial knot and $Z(O)$ its Kontsevich invariant in the source's completed diagram algebra. Its normalization and framing convention are part of the data; one must not automatically replace $Z(O)$ by $1$. Let $\iota$ be the diagram-to-arrow-diagram map referenced in Conjecture 2.16, with the intended extension to this completed setting.

The right-hand side is a closure of a formal exponential with coefficient $1/2$ and a difference of two pictured diagrams. Label those missing pictures $D_1,D_2$, solely as placeholders. The recoverable structure of the conjecture is \[ \iota(Z(O))= \operatorname{cl}\left(\exp\left(\tfrac12(D_1-D_2)\right)\right). \] In a specified complete characteristic-zero diagram algebra and for a positive-degree argument, the formal exponential means \[ \exp\left(\tfrac12(D_1-D_2)\right) =\sum_{k\ge0}\frac{(D_1-D_2)^k}{2^k k!}, \] using the source's product. Completion makes this a degreewise formal expression, not a numerical analytic-convergence claim.

The extraction contains neither $D_1,D_2$ nor the precise map/product/closure operations. The placeholders are not arbitrary diagrams whose choice could be optimized to make the equation true. Closing a skeleton and taking topological closure are also different operations; the source's intended closure must be recovered. Since Conjecture 2.16 was itself pictured, its interval/circle and coefficient conventions cannot be inferred here.

Thus the task is to prove the specific diagram identity of Bar-Natan--Haviv once those missing pictures and conventions are restored, not merely an ordinary formula for the unknot's Kontsevich invariant. The supplied research summary identifies the conjecture but supplies no resolved-status theorem or complete formula.

Short English statement

Prove the stated identity for the arrow-diagram image of the unknot's Kontsevich invariant. Its two defining diagrams, closure/product operations, map extension, and normalization are missing from the extracted formula.

Sources

• Local UnsolvedMath registry, permanent ID $10400058$, exact source code catalogue entry, original Conjecture 3.8, attributed to D. Bar-Natan and A. Haviv.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 3.8, recorded PDF page 55; https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access 2026-07-29; pdf-font-structure extraction.
• Source cross-reference Conjecture 2.16 for $\iota$. The Kontsevich-invariant citation is extracted as "[3 5]"; whether this is [35] or separate references cannot be recovered from the record.