Completeness of Unzipped-Link Invariants for Spatial Theta Graphs

Definitions and mathematical statement

Let $\Theta$ be the graph with two vertices and three edges joining them. A spatial theta-graph is an embedded copy $\Gamma\subset S^3$ of $\Theta$. Temporarily index its edges by $i\in\{1,2,3\}$. Deleting the interior of edge $i$ leaves an embedded circle, whose knot type is denoted by $K_i(\Gamma)$; these are the three constituent knots.

For each edge $i$, write $L_i^n(\Gamma)$ for the two-component link described in the supplied record as obtained by unzipping that edge with linking number $n$, where $n\in{\mathbb{Z}}$. The intended infinite collection is \[ L_\Gamma=\left(K_1(\Gamma),K_2(\Gamma),K_3(\Gamma);\ (L_i^n(\Gamma))_{i\in\{1,2,3\},\,n\in{\mathbb{Z}}}\right), \] with entries regarded as knot or link isotopy types. This display records the intended entries; it does not settle whether the source regards them as an indexed family, a multiset, or a set. The supplied excerpt also does not give a self-contained definition of unzipping or specify the orientations, framing choices, and indexing conventions needed to make $L_i^n(\Gamma)$ unambiguous.

Question, subject to these unresolved conventions. Is $L_\Gamma$ a complete invariant of spatial theta-graphs up to ambient isotopy in $S^3$? More explicitly, once the category of embeddings and the conventions for the collection and its equality have been fixed, does \[ L_\Gamma=L_{\Gamma'} \quad\Longrightarrow\quad \Gamma\text{ and }\Gamma'\text{ are ambient isotopic in }S^3 \] hold for every pair of spatial theta-graphs $\Gamma,\Gamma'$ in that category? Here ambient isotopy means an isotopy $H_t:S^3\to S^3$, with $H_0=\operatorname{id}_{S^3}$ and $H_1(\Gamma)=\Gamma'$; whether it must preserve edge labels, vertex labels, or orientations remains unspecified. Which subcollections still determine the graph, and in what sense can such a subcollection be minimal?

Supplied motivation and its limitation. The excerpt considers the free ${\mathbb{Q}}[a,a^{-1}]$-module generated by links in $S^3$, quotiented by a Conway skein relation, and calls the result $K_{\nabla}$. It describes this as an extended Alexander skein module because no value for the unknot is imposed. It reports that consecutive unzipped links differ by one crossing change and that smoothing the relevant crossing gives $K_i$, up to isotopy after untwisting. In its notation it records \[ [L_i^n]-[L_i^{n-1}]=a[K_i]\qquad(n\in{\mathbb{Z}}). \] The crossing diagrams in the supplied transcription are corrupted, so the precise skein, sign, and orientation conventions cannot be recovered from it. These are reported module relations, not assertions that the corresponding links are isotopic. The excerpt suggests that recording the unzipped links at one fixed value of $n$, for example $n=0$, might suffice, but explicitly says that no proof had been obtained. It does not specify whether that proposed reduced data also retain all three constituent knots. The formulation above therefore remains a statement-only draft with unresolved definition and equivalence conventions.

Short English statement

Does the collection of the three constituent knots and all links obtained by unzipping edges of a spatial theta-graph determine the graph up to ambient isotopy? If so, how much of that collection is necessary? The supplied excerpt suggests considering one fixed unzipping parameter, but gives no proof that this suffices. Precise unzipping, labeling, orientation, and minimality conventions remain unresolved.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006051$, source problem code catalogue entry, title: Completeness of Unzipped-Link Invariants for Spatial Theta Graphs. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied primary citation: MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation (2025), Oberwolfach Reports, pp. 2511--2550, DOI 10.4171/owr/2024/43. https://doi.org/10.4171/owr/2024/43. The transcribed excerpt contains the page marker 2519 and the heading Invariants and Structures in Low-Dimensional Topology. The supplied citation's 2025 date differs from the 2024 report identification; this discrepancy is unresolved here.
• Supplied literature source: Oberwolfach Report 43 (2024). https://doi.org/10.4171/owr/2024/43. Literature checked at: 2026-08-22. This is supplied dated curation, not an independently verified assessment. It labels the problem open, reports that the primary report poses completeness as unanswered and supplies no proof, and states that no later resolution was found in that curation. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. These are supplied provenance claims; no independent source visit or verification is asserted in this draft.