Topological complexity (within 1) of the space of isometry classes of planar n-gons for sufficiently large n

TL;DR

This paper investigates the topological complexity of spaces of planar n-gons classified by genetic codes, establishing bounds for large n and providing evidence for the typical values of TC.

Contribution

It proves bounds on the topological complexity of these spaces based on genetic code structures for large n.

Findings

TC is either 2n-5 or 2n-6 for large n and specific genetic codes.

The largest gee size influences the TC bounds.

It is rare for TC to deviate from these bounds.

Abstract

Hausmann and Rodriguez classified spaces of isometry classes of planar n-gons according to their genetic code, which is a collection of sets (called genes) containing n. Omitting the n yields what we call gees. We prove that, for a set of gees with largest gee of size k>0, the topological complexity (TC) of the associated space of n-gons is either 2n-5 or 2n-6 if n>2k+2. We present evidence that suggests that it is very rare that the TC is not equal to 2n-5 or 2n-6.