Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

Abstract

In a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry ${\mathbb{L}}^3$, ${\mathbb{R}}^3$, ${\mathbb{S}}^3$, ${\mathbb{L}}^2\times \mathbb{R}$, ${\mathbb{S}}^2\times \mathbb{R}$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $\hat{I}“$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,\hat{I}“)$ with an involution $\ddot{I}„$ such that $N(P,\hat{I}“)/⟨\ddot{I}„⟩\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,\hat{I}“)\to S^3$ is a link $C_{\hat{I}}“\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $\hat{I}“$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,\hat{I}“)$. We give a criterion when $S^3\setminus C_{\hat{I}}“$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_{\hat{I}}“$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in ${\mathbb{L}}^3$ with $0$, $2$ or $4$ finite vertices. We give a criterion when the link $C_{\hat{I}}“$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

31 pages, 14 figures. Criterion when a link is hyperbolic is added