Angled decompositions of arborescent link complements

Hyperbolicity of arborescent tangles and arborescent links

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a rational tangle, T=Q_m * T' for some m greater than or equal to 1, or T contains Qn for some n greater than or equal to 2. We use these results to prove a theorem of Bonahon and Seibenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q2.

Embeddability of right-angled Artin groups on complements of trees

For a finite simplicial graph $Γ$, let $A(Γ)$ denote the right-angled Artin group on $Γ$. Recently Kim and Koberda introduced the extension graph $Γ^e$ for $Γ$, and established the Extension Graph Theorem: for finite simplicial graphs $Γ_1$ and $Γ_2$ if $Γ_1$ embeds into $Γ_2^e$ as an induced subgraph then $A(Γ_1)$ embeds into $A(Γ_2)$. In this article we show that the converse of this theorem does not hold for the case $Γ_1$ is the complement of a tree and for the case $Γ_2$ is the complement of a path graph.

Right-angled polyhedra and alternating links

To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.

Hyperbolicity of Augmented Links in the Thickened Torus

For a hyperbolic link K in the thickened torus with no bigons, we show that there is a decomposition of the complement of a link L, obtained from augmenting K, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [5], this shows that L is hyperbolic.

加厚曲面中链环的算术性与可公度性

Arithmeticity and commensurability of links in thickened surfaces

本文针对由常曲率曲面正则4价平铺构建的直角平铺链环，利用对称性论证等给出其算术性的完整刻画，结合典范分解与数论数据确定两两可公度条件。

Hyperbolic three-manifolds that embed geodesically

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.