A remark on the space of 7-gons with a fixed total length in $\R^3$

TL;DR

This paper proves that in the space of 7-gons in three-dimensional space, there exists an isometric isotopy that transforms any polygon into its mirror image while preserving certain projections, using elementary Cayley number arguments.

Contribution

It establishes a novel isometric isotopy in the space of 7-gons that preserves projections and transforms polygons into their mirror images, extending previous models.

Findings

Existence of an isometric isotopy for 7-gons in R^3.

The isotopy transforms polygons into their mirror images.

Preserves lengths of projections onto coordinate planes and lines.

Abstract

Based on the model of the space $Pol_3(n)$ of polygons in $R^3$ with limited number of vertex, which was proposed by Jean-Claude Hausmann and Allen Knutson, and developed by several authors: Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, and Clayton Shonkwiler, we prove that there exists an isometric isotopy of $Pol_3(n)$, $n=7$, into itself, which transforms an arbitrary polygon to its mirror copy, and, additionally, preserves lengths of projections of polygons into the two coordinate planes, and keeps projection of polygons onto the line. The proof is based on elementary arguments with Cayley numbers. A possible generalization of the statement for greater $n$ is related with a theorem by I.James on strong Kervaire invariants in stable homotopy of spheres.