Knot Embeddings in Improper Foldings

TL;DR

This paper introduces the fold number, a new invariant measuring the minimum creases needed to produce any knot via origami foldings, and explores its properties and limitations.

Contribution

It develops the fold number invariant, relates it to diagram stick number, and characterizes foldings that can or cannot produce nontrivial knots.

Findings

All tame knots can be formed by origami foldings with self-intersection.

The fold number is bounded by the diagram stick number.

Proper foldings cannot produce nontrivial knots.

Abstract

Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new knot invariant, the fold number, defined as the minimum number of creases required to obtain an equivalent knot. We study this invariant, presenting a bound on the fold number by the diagram stick number as well as a class of torus knots with constant fold number. We also pursue a characterization of those foldings which admit nontrivial knots, giving a proof that no "physically realizable", or proper, foldings can admit nontrivial knots. A number of questions are posed for further study.