Inscriptions of Isosceles Trapezoids in Jordan Curves

TL;DR

This paper develops a new Lagrangian Floer homology for smooth Jordan curves to prove they inscribe all isosceles trapezoids, extending previous work and including non-smooth cases.

Contribution

It introduces a novel Floer homology framework that generalizes previous results and applies to non-smooth Jordan curves for inscribing isosceles trapezoids.

Findings

Every smooth Jordan curve inscribes every isosceles trapezoid.

The new homology re-establishes known results and extends them to non-smooth curves.

Spectral invariants help identify non-smooth curves that also inscribe isosceles trapezoids.

Abstract

We construct a Lagrangian Floer homology whose chain complex is generically generated by the inscriptions of isosceles trapezoids in a smooth Jordan curve. This is an extension of Greene and Lobb's Jordan Floer homology (arXiv:2404.05179), which we also call Jordan Floer homology. Its non-triviality re-establishes that every smooth Jordan curve inscribes every isosceles trapezoid. By consideration of the spectral invariants associated with the real filtration known as the action filtration, we establish new cases of non-smooth Jordan curves which admit inscriptions of isosceles trapezoids.