Inscribed Squares and Relation Avoiding Paths

TL;DR

This paper explores the link between inscribed squares in Jordan curves and relation avoiding paths in complex vector spaces, proposing conjectures that connect these geometric and topological problems.

Contribution

It introduces a novel connection between inscribed square problems and relation avoiding paths, and formulates conjectures relating these concepts in complex vector spaces.

Findings

A Jordan curve without inscribed squares would have a complex structure called a square envelope.

Conjectures about relation avoiding paths imply inscribed squares in Jordan curves with singularities.

The work suggests a deep link between geometric configurations and topological path properties.

Abstract

We develop a connection between the inscribed square problem and the question of understanding relation avoiding paths in a complex vector space. Our main theorem is that a Jordan curve with no inscribed squares would have a seemingly impossible structure which we call a square envelope. We will make some conjectures about the nature of relation avoiding paths in vector spaces, and show that these conjectures would imply the existence of inscribed squares in Jordan curves with finitely many arbitrarily complicated singularities.