Configuration Spaces of Tensegrities — Problem 7

Definitions and mathematical statement

For a graph $G=(\{1,\ldots,n\},E)$ and a planar placement $p=(p_1,\ldots,p_n)$, an equilibrium self-stress is a real edge weighting $w=(w_{ij})_{\{i,j\}\in E}$ satisfying \[ \sum_{j:\{i,j\}\in E}w_{ij}(p_j-p_i)=0\qquad(i=1,\ldots,n). \] These weightings form a vector space, and their coordinate signs distinguish tension/compression patterns. The source uses the tensegrity stratification of the placement base $B_2(K_n)$, but does not give its exact fiber/sign-equivalence definition or the permitted degenerate placements.

Does there exist a classical Cayley-algebra system, or several such systems, describing precisely the union of the codimension-one tensegrity strata in the plane? The source opens with an arbitrary graph $G$ but identifies the target parenthetically as codimension-one strata of $B_2(K_n)$; the relation between these two graph choices is not supplied.

Cayley algebra here refers to projective/Grassmann--Cayley incidence constructions, not the octonion algebra. For example, planar points can be homogenized to $(x_i,y_i,1)\in\mathbb R^3$ and incidence expressed through exterior-algebra joins/meets and determinant brackets. The strict requested description must use only the original configuration elements, without the auxiliary elements admitted by an extended Cayley algebra. The exact permitted expressions, equations/nonvanishing conditions, and interpretation of several systems remain unspecified.

The supplied source reports a weaker extended-Cayley solution in Karpenkov (2017), while explicitly leaving removal of its additional elements unresolved. It also refers to Problem 8 as a concrete candidate counterexample to the strict version; that example is not included and is not reconstructed. Neither the relaxed result nor an arbitrary coordinate-polynomial description is treated as the requested strict Cayley description.

Short English statement

Describe exactly the union of planar codimension-one tensegrity strata using classical Cayley-algebra systems with no auxiliary configuration elements. The extended-algebra version is reported solved, while the strict version and some target conventions remain unresolved.

Sources

• Oleg Karpenkov, Open Problems on Configuration Spaces of Tensegrities (2020), Problem 7, PDF page 5, https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf; accessed 2026-07-29, PDF-text-normalized extraction. Problem 8 is mentioned as a candidate strict-version counterexample.
• Karpenkov (2017), arXiv:1512.02563, as the supplied extended-Cayley solution reference; the record gives no separate article title.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6900007$, source code catalogue entry, Configuration Spaces of Tensegrities --- Problem 7. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.