Configuration Spaces of Tensegrities — Problem 2

Definitions and mathematical statement

The supplied question asks for all different types of strata for ten points, in the planar tensegrity-configuration setting described by the research summary. The source excerpt does not provide the definition of the configuration space or its stratification.

For context, for a graph $G$ on labeled vertices $1,\ldots,10$ and positions $p=(p_1,\ldots,p_{10})\in(\mathbb R^2)^{10}$, an equilibrium stress is an assignment $\omega_{ij}=\omega_{ji}\in\mathbb R$ to its edges satisfying \[ \sum_{j:\{i,j\}\in E(G)}\omega_{ij}(p_j-p_i)=0 \qquad(1\le i\le10). \] These assignments form a linear space $W(G,p)$. Tensegrity stratifications can track more than its dimension, for instance sign data or sign-preserving equivalence of fibers. The question is to classify every stratum type in the source's ten-point space, not merely possible stress dimensions or codimension-one strata.

The graph, whether all pairs are edges, labeled-versus-unlabeled convention, admissible point coincidences/degeneracies, any geometric quotient, and precise equivalence defining a stratum "type" are missing. The displayed stress space gives standard context without choosing those absent ingredients. In particular a rank stratification alone is not asserted to be the intended tensegrity stratification.

The supplied summary quotes the source's progress: codimension-one strata for $n\le8$ were classified by Doray and collaborators (2010), and Karpenkov (2017) gave an approach to every stratum for $n=9$; it identifies $n=10$ as the next case containing unknown strata. "An approach" is not silently upgraded to a complete lower-dimensional classification. The record reports the ten-point problem open, without an independently verified modern resolution.

Short English statement

Classify every stratum type in the tensegrity configuration space for ten planar points. The source's graph, admissible configurations, and definition of type are missing; the cited smaller-case progress concerns codimension-one classification and an approach to the nine-point case.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6900002$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Oleg Karpenkov, Open Problems on Configuration Spaces of Tensegrities (2020), Problem 2, PDF page 3; progress discussed in section 3: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29.
• Doray and collaborators (2010), cited by the supplied summary for codimension-one strata with $n\le8$; no full title or bibliographic details are supplied.
• Karpenkov (2017), arXiv:1512.02563, cited for the nine-point approach; no full title is supplied in the record.