The Crisscross and the Cup: Two Short 3-Twist Paper Moebius Bands

TL;DR

This paper introduces two specific 3-twist polygonal paper Moebius bands, the crisscross and the cup, and explores their properties and limits in relation to knotted boundary conditions.

Contribution

It presents the first explicit constructions of these two 3-twist paper Moebius bands and analyzes their role as limits of smooth embedded bands with knotted boundaries.

Findings

The crisscross is planar, while the cup is not.

Both objects are limits of smooth embedded bands with knotted boundary.

A conjecture relates aspect ratio to knotted boundary complexity.

Abstract

We introduce the crisscross and the cup, both of which are immersed $3$-twist polygonal paper Moebius band of aspect ratio $3$. We explain why these two objects are limits of smooth embedded paper Moebius bands having knotted boundary. We conjecture that any smooth embedded paper Moebius band with knotted boundary has aspect ratio greater than $3$. The crisscross is planar but the cup is not.