Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to $\sqrt{3}$

TL;DR

The paper proves that every smooth Jordan curve inscribes a rectangle with a specific aspect ratio, using knot theory bounds, notably establishing the case where the aspect ratio is 3.

Contribution

It introduces a novel application of knot theory to the inscribed rectangle problem, establishing the existence of rectangles with particular aspect ratios.

Findings

Every smooth Jordan curve inscribes a rectangle with aspect ratio 3.

The method applies Batson's lower bound on nonorientable slice genus.

The result confirms the inscribed rectangle conjecture for aspect ratio 3.

Abstract

We use Batson's lower bound on the nonorientable slice genus of $(2n,2n-1)$-torus knots to prove that for any $n \geq 2$, every smooth Jordan curve has an inscribed rectangle of of aspect ratio $\tan(\frac{\pi k}{2n})$ for some $k\in \{1,...,n-1\}$. Setting $n = 3$, we have that every smooth Jordan curve has an inscribed rectangle of aspect ratio $\sqrt{3}$.