Knotted Hamiltonian cycles in linear embedding of $K_7$ into $\mathbb{R}^3$

TL;DR

This paper investigates the presence of knotted Hamiltonian cycles in linear embeddings of the complete graph K_7 in three-dimensional space, establishing an upper bound on the number of figure-8 knots such cycles can contain.

Contribution

It proves that any linear embedding of K_7 contains at most three figure-8 knots as Hamiltonian cycles, providing a new bound in the study of knots in graph embeddings.

Findings

Maximum of three figure-8 knots in any linear embedding of K_7

Extension of knot theory in graph embeddings

Insight into intrinsic knot properties of K_7

Abstract

In 1983 Conway and Gordon proved that any embedding of the complete graph $K_7$ into $\mathbb{R}^3$ contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper we are interested in knotted Hamiltonian cycles in linear embedding of $K_7$. Concretely it is shown that any linear embedding of $K_7$ contains at most three figure-8 knots as its Hamiltonian cycles.