Alejandro Ucan-Puc
In this paper we present a classical construction of the Hyperbolic structure of the complement of a link in the sense of Thurston for the particular case of the Borromean rings link. As this is nothing new, the aim of this paper is to complete the literature about this topic in the particular case of the Borromean rings complement.
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In this note, we show the fundamental group of the complement of the Borromean rings in $\Bbb{S}^3$ has exactly two representations in ${\rm PSL}(2,\Bbb{C})$ which are faithful, discrete and send meridians into parabolic elements. Using this result we are able to show that Borromean rings admit exactly one hyperbolic structure.
The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.
Canonical triangulations of Dehn fillings of the Borromean rings link complement
本文为 Borromean 环链补空间及其相关流形的 Dehn 填充构造典范三角剖分，通过证明每个面的局部凸性，扩展了已知典范分解的无限族。
On the volume conjecture of the colored Jones invariants with arbitrary colors
研究着色Jones不变量在链接补集超几何结构变形下的体积猜想，通过图八结和布洛美环的极限展示与超几何锥流形体积的关系。
Profinite rigidity and geometric convergence
本文证明 profinitely rigid 有限体积双曲流形在几何拓扑下闭合并借助 bubble-drilling 构造推广到一大类带尖点双曲流形，涵盖 Whitehead link 与 Borromean ring 补空间。
Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in $S^4$
该研究针对四维球面中二维环面的双曲链环补空间，证明其内部不存在闭的嵌入全测地双曲3-流形，为高维双曲流形的几何结构研究提供了新结论。
We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.
We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$. Our works complement the work of Lakeland and Leininger \cite{Christopher} on the upper bound of systole length.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to a standard $S^2 \times S^2$.
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311.
We construct examples of codimension two hyperbolic link complements in closed smooth 4-manifolds with homeomorphism type $\#_{2k}S^2 \times S^2$. All our examples are based on a construction of J. Ratcliffe and S. Tschantz, who constructed 1171 non-compact finite volume hyperbolic 4-manifolds of minimal volume. We then give necessary conditions for a closed smooth simply connected 4-manifold to contain a codimension two link complement that admits a hyperbolic structure.
We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures.
We establish a pair of criteria for proving that most knot complements obtained as Dehn fillings of a given two-component hyperbolic link complement lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite families of knot complements, all but finitely many members lack hidden symmetries.
An alternative method is described for determining the hyperbolic structure on a link complement, and some of its elementary consequences are examined. The method is particularly suited to alternating links.
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$ shortest closed geodesic of $M\setminus L$ in terms of $V$.
It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.
Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.
This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a $S^2(2,3,6)$ cusp, it also covers an orbifold with a $S^2(3,3,3)$ cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.
Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $\mathbb S^3$. Then, if $M$ admits an exhaustion by $π_1$-injective sub-manifolds there exists cantor sets $C_n\subset \mathbb S^3$ such that $N_n=\mathbb S^3\setminus C_n$ is hyperbolic and $N_n\rightarrow M$ geometrically.
Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some of those manifolds are then shown to be complements of tori and Klein bottles in other simply-connected closed 4-manifolds. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic 4-manifolds of minimal volume. Our examples a…
Hyperbolic Handlebody Complements in 3-Manifolds
本文证明紧致可定向3-流形经边界处理后包含任意亏格柄体且其补空间闭包为双曲流形，并通过扩展八面体分解得到部分柄体补的体积界限。
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Our work requires a carefu…
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.
Volume bounds for hyperbolic rod complements in the 3-torus
该研究针对3-环面中双曲杆补的几何性质，依据杆参数给出体积的上下界，还针对一类杆补提出了仅依赖连分数长度的更优渐近尖锐体积界。
In this paper, we study the generalized volume conjecture for the colored Jones polynomials of links with complements containing more than one hyperbolic piece. First of all, we construct an infinite family of prime links by considering the cabling on the figure eight knot by the Whitehead chains. The complement of these links consist of two hyperbolic pieces in their JSJ decompositions. We show that at the $(N+\frac{1}{2})$-th root of unity, the exponential growth rates for the $\vec{N}$-th colored Jones polynomials for these links capture the simplicial volume of the link complements. As an …
For any g>1 we construct a graph G_g in S^3 whose exterior M_g supports a complete finite-volume hyperbolic structure with one toric cusp and a connected geodesic boundary of genus g. We compute the canonical decomposition and the isometry group of M_g, showing in particular that any self-homeomorphism of M_g extends to a self-homeomorphism of the pair (S^3,G_g), and that G_g is chiral. Building on a result of Lackenby we also show that any non-meridinal Dehn filling of M_g is hyperbolic, thus getting an infinite family of graphs in S^2xS^1 whose exteriors support a hyperbolic structure with g…
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which …
Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to having infinitely many fillings of a cusped manifold produce knot complements admitting hidden symmetries. Applying these tools, we show for any two-bridge link complement, at most finitely many fillings of one cusp can be covered by knot complements admitting hidden symmetries. We also show that the figure-eight knot complement is the unique knot complement with…
This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifolds with the same property. We also show that the two-component link complements above have integral traces, but each has a mutant with a nonintegral trace.
On a family of hyperbolic Brunnian links and their volumes
本文构造推广 Debrunner 例子的一族多分支 Brunnian 链环，并利用带尖点流形的 Dehn 填充给出其补空间双曲体积上界。
Volume Conjecture and quantum hyperbolic invariants: the figure eight knot complement
研究八边形结补集的量子双曲不变量的半经典极限实部，证明其与缠绕 longitude 上的 holonomy 特征值对数的奇偶条件相关，贡献在于揭示其与双曲体积的关系。
We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirting…
This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published explanations of SnapPea's algorithms for triangulating a link complement efficiently and for converging quickly to the hyperbolic structure while avoiding singularities in the parameter space.
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.
We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.
One-cusped Dehn fillings of the sisters of the Whitehead and $6^2_2$ link complements
研究（-2,3,8）-椒盐卷饼链补空间和伯格流形单尖点德恩填充的算术性，通过特定技术表明其尖点域等重合，伯格流形无算术性，（-2,3,8）-椒盐卷饼链补空间唯一算术填充是八字结补空间姐妹空间，且相关纽结补无隐藏对称性。
If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the complement of the graph bounds the graph distance of the bridge surface. We use this result to construct, for any natural number $n$, a hyperbolic manifold containing a surface of topological index $n$.
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.
We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections.
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.
Given a compact orientable 3-manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C is as small as you like.
We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the invariant trace field of the link. On the other hand, Yokota suggested a potential function of a hyperbolic knot, which gives the hyperbolicity equations and the complex volume of the knot. Applying the fact above to his theory, we explain that the potential function also gives all …
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting handlebodies has positive genus, and gluing a pair of pieces together along the twice-punctured disks in their boundaries, we show the result is also hyperbolic. This should be contrasted with composition of any pair of knots in the 3-sphere, which is never hyperbolic. Similar results are obtained when both twice-punctured disks are in the same handlebody and we glue …
In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in $PSp(2,1)$ out of PU(2,1) up to conjugacy.
We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.
We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: - for every integer n we can prove that there are infinitely many hyperbolic knots in the 3-sphere having exceptional surgeries n, n+1, n+2, n+3, with n+1, n+2 giving small S…
We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibered with the same genus and are chiral. A characterisation of cyclic commensurability classes of complements of periodic knots is also given. In the non-periodic case, we reduce the characterisat…
This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.
By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction. In the case of a classical…
Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in $S\times I$ is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in $S\times I$. A fully alternating link is prime in $S\times I$ if and only if it is "obviously prime". Fur…
This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.
We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
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