A Jordan Curve that cannot be Crossed by Rectifiable Arcs on a Set of Zero Length

TL;DR

This paper constructs a specific Jordan curve in the complex plane that prevents rectifiable arcs connecting different regions from intersecting it in a set of positive length, revealing a unique geometric property.

Contribution

It introduces a Jordan curve with a novel property that any rectifiable arc connecting different sides must intersect it in a set of positive length, which was previously unknown.

Findings

Constructed a Jordan curve with this unique intersection property

Proved that any rectifiable arc connecting different components intersects the curve in positive length

Demonstrated the existence of such a curve in the complex plane

Abstract

We construct a Jordan curve $\Gamma \subset \mathbb{C}$ so that for any rectifiable arc $\sigma$ with endpoints in distinct complementary components of $\Gamma$, $H^1(\sigma \cap \Gamma) > 0$.