Heptagonal knots and Radon partitions

TL;DR

This paper characterizes when a heptagonal knot is a figure-8 knot using Radon partitions and relates this to the count of nontrivial heptagonal knots in linear embeddings of K_7.

Contribution

It provides a necessary and sufficient condition for heptagonal knots to be figure-8 knots based on Radon partitions, linking knot theory and geometric combinatorics.

Findings

Radon partitions characterize figure-8 knots among heptagonal knots.

The paper establishes a link between knot classification and embeddings of K_7.

Results inform the enumeration of nontrivial heptagonal knots in specific embeddings.

Abstract

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.