5 problems

Let $P$ be a convex $n$-gon, and place $n$ distinct points on the moment curve in $\mathbb{R}^{4}$. The associated bar-and-joint rigidity matroid is represented by the rigidity mat…

Let $P$ be a convex $n$-gon, and let $k$-triangulations be its maximal sets of diagonals containing no $k+1$ mutually crossing diagonals. Place $n$ generic points on the moment cur…

Arbitrary-position rigidity conjecture. Reduced bipartized $2$-triangulations are bases in the bipartite bar-and-joint rigidity matroid of these points.

Rigidity realization conjecture. Reduced bipartized $k$-triangulations of the $n$-gon are bases in the bar-and-joint rigidity matroid of these generic points.

Jonsson's polytopality conjecture. For every $n\geq 2k+1$, the complex $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$ is a polytopal sphere: there is a simplicial polytope of…