Kirby Problem 1.72

Definitions and mathematical statement

An equilateral spatial $n$-gon is a cyclic sequence of vertices $v_1,\ldots,v_n\in\mathbb R^3$, with indices modulo $n$, such that $|v_{i+1}-v_i|=1$. Require the resulting closed polygonal curve to be embedded: nonadjacent edges are disjoint, and adjacent edges meet only at their common endpoint. It is unknotted if this curve is ambiently isotopic to a round circle.

A permitted deformation is a continuous family of vertex positions $v_i(t)$, $0\leq t\leq1$, that remains embedded and retains all $n$ edge lengths equal to one. The vertices and their cyclic order persist throughout; adding vertices, passing edges through one another, and changing edge lengths are prohibited. Call the initial polygon stuck if no such deformation ends at a convex planar polygon.

Is there an integer $n$ and an unknotted equilateral embedded $n$-gon that is stuck? Equivalently, does some equilateral unknot lie in a path component of this polygon configuration space containing no convex planar configuration? Fixing unit length loses no generality, because simultaneous scaling converts any common positive edge length into one.

Short English statement

Can an unknotted polygon with all edges equal be impossible to unfold into a convex planar polygon while preserving edge lengths?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2731 (KP-1.72), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.72. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.