Question 2.4 — (T.

Definitions and mathematical statement

Let $X=\downarrow\!\downarrow$ be the source's two oriented interval strands and let $A(X;{\mathbb{Z}})$, $A(S^1;{\mathbb{Z}})$ be its integral diagram modules. They consist of integer linear combinations of diagrams on the indicated skeletons modulo the specified diagram relations. The exact chord/Jacobi, framing, and relation conventions are not included here and must not be changed when testing torsion.

Write $\tau\in A(X;{\mathbb{Z}})$ for the Dogolazky--Kneissler element shown in the source's Figure 7, reported to satisfy \[ \tau\ne0,\qquad2\tau=0. \] For each admissible diagram embedding/insertion $e$ into a circle-skeleton diagram, let $j_e$ denote the induced integral-linear operation, and put $x_e=j_e(\tau)\in A(S^1;{\mathbb{Z}})$. Since the operation respects the relations, \[ 2x_e=j_e(2\tau)=0. \] Stanford's precise nonvanishing target is \[ \exists e\text{ admissible},\qquad j_e(\tau)\ne0. \] Such an image would have exact order two. Nonzero source torsion does not itself imply a nonzero image, since the insertion/closure operation need not be injective.

The source says a positive answer would refute Conjecture 2.3, described in the supplied summary as torsion-freeness of $A(S^1;{\mathbb{Z}})$. That condition is \[ \forall x\in A(S^1;{\mathbb{Z}})\ \forall m\ge1,\qquad mx=0\Longrightarrow x=0. \] Conversely, vanishing of all images of this one element would not alone establish torsion-freeness for every circle diagram.

Figure 7's actual integral combination and the complete set of allowed embeddings/auxiliary diagram insertions are absent, so no concrete image or exhaustive test can be reconstructed. The trailing text "2.2 Do Vassiliev invariants distinguish knots?" is an adjacent section heading, not another clause of Question 2.4. The supplied research summary reports no known nontrivial image, without a verified exhaustive result.

Short English statement

Can some allowed embedding of the specified two-strand order-two diagram element give a nonzero circle-diagram class? Its image is automatically killed by two, but nonvanishing needs the missing Figure 7 and exact integral diagram conventions.

Sources

• Local UnsolvedMath registry, permanent ID $10400026$, exact source code catalogue entry, original Question 2.4, attributed to T. Stanford.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Question 2.4, recorded PDF page 29; https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access 2026-07-29; pdf-font-structure extraction.
• Source Figure 7 for the Dogolazky--Kneissler element and Conjecture 2.3 for the circle-module torsion claim; the figure and full convention definitions are not supplied.
• Supplied OPEN-TRIAGE summary reports no known nontrivial embedded image, with no additional precise publication citation.