Determining Knotsw and Links by Cyclic Branched Coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD₆-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.     

On $ \pi $-hyperbolic knots and branched coverings

We prove that, for any given , a -hyperbolic knot is determined by its 2-fold and n-fold cyclic branched coverings. We also prove that a -hyperbolic knot which is not determined by its m-fold and n-fold cyclic branched coverings, , must have genus . Received: December 14, 1998.     

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. Mathematics Subject Classification (2000):57N10, 57M25.The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Isometry Groups of Hyperbolic Three-Manifolds which are Cyclic Branched Coverings

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different -hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different -hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Thin position for knots and 3-manifolds

We prove that for 2-bridge knots and 3-bridge knots in thin position the double branched cover inherits a manifold decomposition in thin position. We also argue that one should not expect this sort of correspondence to hold in general.     

Conjugacy problem in groups of oriented geometrizable 3-manifolds

The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u,v of π₁(M) whether u and v are in the same conjugacy class of π₁(M) or not. More topologically, the algorithm decides for any two loops in M, whether they are freely homotopic or not.     

Behavior of the extended complexity of irreducible 3-manifolds

We construct the extended complexity of irreducible 3-manifolds; unlike the usual complexity [1] it is not an integer, but an ordered tuple of five integers. The benefit of extended complexity is that it always decreases when a manifold is cut along some incompressible boundary incompressible surface.     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

Incompressible Surfaces in Handlebodies and Closed 3-manifolds

in this paper we prove that for any positive integer n, 1) a handlebody of genus 2contains a separating incompressible surface of genus n, and 2) there exists a closed 3manifold of heegaard genus 2 which contains a separating incompressible surface of genus n.     

Edges of Canonical Decompositions for 2-Bridge Knots and Links

Let L=C(a1,..., ak) be a hyperbolic 2-bridge knot or link. We give a family of arcs in S3–L which are ambient isotopic to edges of the canonical decomposition of S3–L if each aj is sufficiently large. This result supports the conjecture that the decomposition of S3–L given by Sakuma–Weeks in [12] is the canonical one.     

A census of genus-two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karábaš, Mali?ký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.     

Diffeomorphism Type of Certain 3-connected Closed Smooth 12-manifolds

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Epstein–Penner Decompositions and Thrice Punctured Spheres in Hyperbolic 3-Manifolds

Let M be a cusped hyperbolic 3-manifold containing an incompressible thrice punctured sphere S. Suppose that M is not the Whitehead link complement. We prove that a certain arc on S is isotopic to an edge of a Euclidean decomposition of M. By using the above result, we relate alternating knot diagrams and the canonical decompositions. Let K be an alternating hyperbolic knot. On a reduced alternating knot diagram of K, if we replace one of the crossings with a large number of half twists, the polar axis of the crossing is isotopic to an edge of the canonical decomposition for the resulting knot.     

Higher dimensional cable knots and their finite cyclic covering spaces

As an analogue of the classical cable knot, the p-cable n-knot about an n-knot K, where p is an integer and n?2, is defined, and some basic properties of higher dimensional cable knots are described. We show that for p>0 then p-fold branched cyclic covering space of an (n+2)-sphere branched over the p-cable knot about an n-knot K is an (n+2)-sphere or a homotopy (n+2)-sphere which is the result of Gluck-surgery on the composition of p copies of K according as if p is odd or even. At the same time, we prove that for any n?2 and p?2, the composition of p copies of any n-knot K is the fixed point set of a Z_p-action on an (n+2)-sphere. This is another counterexample to the higher dimensional Smith conjecture.