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

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.