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

TL;DR

This paper explores hyperbolic links associated with Hamiltonian subgraphs in simple 3-polytopes, providing criteria for hyperbolic structures, classifying such links, and connecting them to quantum mechanics phenomena.

Contribution

It generalizes previous constructions to simple 3-polytopes, offers criteria for hyperbolic structures, and classifies hyperbolic links via Eulerian cycles and subgraphs.

Findings

Hyperbolic links are parametrized by nonselfcrossing Eulerian cycles and subgraphs.

Criteria are provided for when the link complement admits a finite volume hyperbolic structure.

Nontrivial unlinked circle links contain the Borromean rings.

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 $\Gamma$ 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.