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

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 $\Gamma$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,\Gamma)$ with an involution $\tau$ such that $N(P,\Gamma)/\langle\tau\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,\Gamma)\to S^3$ is a link $C_\Gamma\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $\Gamma$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,\Gamma)$. We give a criterion when $S^3\setminus C_\Gamma$ 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 $C_\Gamma$ 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 $C_\Gamma$ 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.