Related papers: Hyperbolic covering knots

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schl\"{a}fli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we…

We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound…

Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element.…

Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is…

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In…

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…

We introduce a numerical invariant \beta(K) of a knot K which measures how non-alternating K is. We prove an inequality between \beta (K) and the (knot Floer) thickness of K. As an application we show that all Montesinos knots have…

The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two…

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite…

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L)…

Given a class $\mathcal{P}$ of groups we say that a group $G$ is fully residually $\mathcal{P}$ if for any finite subset $F$ of $G$, there exists an epimorphism from $G$ to a group in $\mathcal{P}$ which is injective on $F$. It is known…

A knot K in the 3-sphere is superslice if there is a slice disk D in the 4-ball such that the double of D along K is the unknotted 2-sphere S in $S^4$. Answering a question of Livingston-Meier, we find smoothly slice (in fact doubly slice)…

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…

Any profinite isomorphism between two cusped finite-volume hyperbolic 3-manifolds carries profinite isomorphisms between their Dehn fillings. With this observation, we prove that some cusped finite-volume hyperbolic 3-manifolds are…

We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves,…

We survey aspects of classical combinatorial sutured manifold theory and show how they can be adapted to study exceptional Dehn fillings and 2-handle additions. As a consequence we show that if a hyperbolic knot $\beta$ in a compact,…

If a torsion-free hyperbolic group G has 1-dimensional boundary, then the boundary is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When the boundary of G is a Sierpinski carpet we show that G is a…

A slope $p/q \in \mathbb{Q}$ is characterising for a knot $K \subset \mathbb{S}^3$ if the oriented homeomorphism type of the manifold $\mathbb{S}^3_K(p/q)$ obtained by Dehn surgery of slope $p/q$ on $K$ uniquely determines the knot $K$. We…