Configuration Spaces of Tensegrities — Problem 8

Definitions and mathematical statement

The source asks for Cayley algebra systems, if they exist, defining the tensegrity strata of "the following graph", and describes that graph as a candidate counterexample to Problem 7. The graph drawing or its vertex/edge list is not included in the supplied record; denote this missing graph by $G_0$. The exact configuration space, stratification and Problem 7 are also absent.

In the usual projective-geometric context, classical Grassmann--Cayley algebra expresses joins and meets of subspaces by exterior-algebra operations. For example, homogeneous planar point vectors have bracket coordinates $[ijk]=\det(p_i,p_j,p_k)$. This contextual meaning is distinct from unrelated uses of "Cayley algebra" and does not determine which expressions or systems are admitted in the source's strict formulation.

The task is to construct systems in that permitted syntax defining the specified strata of $G_0$, or establish that such systems do not exist. The excerpt does not state whether exact strata, their closures, or a union of codimension-one strata must be defined, nor whether equations alone, inequalities, order conditions, or component data are allowed. These distinctions cannot be supplied by an arbitrary polynomial encoding.

The research summary says the example is related to a six-points-on-a-conic condition and a $K_{3,3}$ strata union, with subtle ordering dependence. It does not identify the missing graph as $K_{3,3}$, so that graph is not substituted for $G_0$. It reports the required systems not known and calls this a strong candidate counterexample, not a proved counterexample. The historical open label and later summary remain source assessments without independent verification.

Short English statement

Find classical Cayley-algebra descriptions of the intended graph's tensegrity strata, or prove none exist in the permitted form. The graph, syntax restrictions, and exact stratum target are missing, and the related conic example is only a candidate obstruction.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6900008$, 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 8, PDF page 5, referring to Problem 7: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29.
• The graph figure and full Problem 7 are not included in the record, and no separate resolution reference is supplied.