Motion Groups of Hopf-Tree Links

Definitions and mathematical statement

Let $\Gamma=(V,E)$ be a graph that is a disjoint union of trees, as in the supplied question. Its right-angled Artin group, with the standard edge-commutation convention, is \[ A_\Gamma=\left\langle x_v\ (v\in V)\ \middle|\ [x_v,x_w]=1\text{ whenever }\{v,w\}\in E\right\rangle, \] where $[a,b]=a^{-1}b^{-1}ab$. The source does not explicitly specify finiteness or whether isolated vertices are included in the intended class of defining graphs.

Write $L_\Gamma\subset{\mathbb{R}}^3$ for a Hopf-tree link corresponding to $\Gamma$ in the sense intended by the source. The supplied record does not define the construction of $L_\Gamma$, including the choices of Hopf linking, component orientations, or how distinct tree components are realized together. Consequently, this notation retains an unresolved source-dependent meaning; pairwise linking data alone are not supplied as a definition of the link.

For a specified configuration or embedding space $\mathcal E(L_\Gamma)$ with base configuration $e_\Gamma$, write \[ \operatorname{Mot}(L_\Gamma)=\pi_1\left(\mathcal E(L_\Gamma),e_\Gamma\right). \] This describes the fundamental-group interpretation of a motion group, conditional on identifying the intended space. The record does not specify whether embeddings are parametrized, whether components are labelled or oriented, what topology and regularity are used, or whether reparametrizations are quotiented out. These choices must be fixed before this expression specifies the intended group.

The phrase symmetric automorphism group also requires a convention. For example, two candidate conventions are \[ \begin{aligned} \mathcal S_+(A_\Gamma) &=\left\{\phi\in\operatorname{Aut}(A_\Gamma): \begin{array}{l} \exists\,\sigma\in\operatorname{Sym}(V),\ \exists\,(g_v)_{v\in V}\in A_\Gamma^V,\\ \forall v\in V,\quad\phi(x_v)=g_vx_{\sigma(v)}g_v^{-1} \end{array}\right\},\\ \mathcal S_{\pm}(A_\Gamma) &=\left\{\phi\in\operatorname{Aut}(A_\Gamma): \begin{array}{l} \exists\,\sigma\in\operatorname{Sym}(V),\ \exists\,(g_v)_{v\in V}\in A_\Gamma^V,\\ \exists\,(\epsilon_v)_{v\in V}\in\{-1,1\}^V,\\ \forall v\in V,\quad\phi(x_v)=g_vx_{\sigma(v)}^{\epsilon_v}g_v^{-1} \end{array}\right\}. \end{aligned} \] Restricting $\sigma$ to the identity gives corresponding pure versions. These are explicitly displayed alternatives, not an identification of the source's intended convention.

Question, with unresolved conventions retained. For every Hopf-tree link $L_\Gamma\subset{\mathbb{R}}^3$ covered by the source, whose defining graph $\Gamma$ is a disjoint union of trees, is there a group isomorphism \[ \operatorname{Mot}(L_\Gamma)\cong\operatorname{SymAut}(A_\Gamma)? \] Here $\operatorname{SymAut}$ denotes the source's intended symmetric automorphism group. A fully determined conjecture requires the missing link construction, motion-space conventions, and automorphism convention. The record asks for an isomorphism; it does not specify a particular homomorphism that must induce it.

Short English statement

The question asks whether the motion group of each intended Hopf-tree link in ${\mathbb{R}}^3$, with defining graph a disjoint union of trees, is isomorphic to the symmetric automorphism group of the associated right-angled Artin group. The supplied record leaves the link construction, embedding-space conventions, and meaning of symmetric automorphisms unspecified, so the displayed question remains conditional on those definitions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004409$, title Motion Groups of Hopf-Tree Links, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2020 and supplies no proposer attribution.
• Supplied source citation: Low-dimensional Topology (2021), Oberwolfach Reports, pp. 465--516, DOI 10.4171/owr/2020/8. https://doi.org/10.4171/owr/2020/8. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labelled Oberwolfach Report 8/2020: https://ems.press/content/serial-article-files/46844.
• Supplied literature source labelled Goldsmith and related motion-group work: https://eprints.whiterose.ac.uk/id/eprint/148587/. No complete bibliographic citation or precise supporting result is supplied for this link.
• Supplied literature source also labelled Oberwolfach Report 8/2020: https://doi.org/10.4171/owr/2020/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that motion groups have been computed for several small link cases, while the proposed identification for defining graphs that are disjoint unions of trees remains conjectural. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The source requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. These are supplied editorial claims, not independent mathematical verification of this draft.