Configuration Spaces of Tensegrities — Problem 1

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a simple graph with labeled vertices $1,\ldots,n$. A configuration is $P=(p_1,\ldots,p_n)\in(\mathbb R^d)^n$; coincident vertices are allowed. A self-stress is a real edge assignment $w_{ij}=w_{ji}$, set to zero on missing edges, satisfying equilibrium at every vertex: \[\sum_{j\ne i}w_{ij}(p_j-p_i)=0\quad(1\leq i\leq n).\] Let $W(G,P)$ be the vector space of all such self-stresses. Negative and positive edge values are the source’s cable and strut signs. The base $B_d(G)$ is the full labeled configuration space $(\mathbb R^d)^n$, stratified by the following fiber equivalence: $W(G,P)$ and $W(G,Q)$ are equivalent if a homeomorphism between them preserves the sign of every edge coordinate of every stress. A stratum is a maximal connected set of configurations with equivalent fibers; these strata are semialgebraic. This base is not quotiented by Euclidean or affine motions. Write $K_n$ for the complete graph on $n$ labeled vertices.

Statement recorded in the supplied source.

Describe the combinatorics of the strata of $B_2(K_6)$, $B_3(K_4)$ and $B_3(K_5)$, in the manner of the explicit stratified descriptions of $B_2(K_4)$ and $B_2(K_5)$ in the source. [S2]

Short English statement

Describe the stratified configuration spaces for six planar vertices and for four or five spatial vertices, including their stratum combinatorics.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6900001 (AMR-068-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Oleg Karpenkov, Jan Schepers and Brigitte Servatius, On stratifications for planar tensegrities with a small number of vertices, arXiv:1201.3557v1 (2012), Definitions 1.1--1.3, pp. 2--3; Problem 2.7, p. 12. https://arxiv.org/pdf/1201.3557.