Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914529570455552 |
|---|---|
| author | Erokhovets, Nikolai |
| author_facet | Erokhovets, Nikolai |
| contents | 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 $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ 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 $C_Γ$ 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 $C_Γ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes Erokhovets, Nikolai Geometric Topology Algebraic Topology Combinatorics 57S12, 57K10, 57K32, 52B10, 57S25, 52B70, 52B05 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 $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ 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 $C_Γ$ 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 $C_Γ$ 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. |
| title | Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes |
| topic | Geometric Topology Algebraic Topology Combinatorics 57S12, 57K10, 57K32, 52B10, 57S25, 52B70, 52B05 |
| url | https://arxiv.org/abs/2512.03017 |