Closed incompressible surfaces in 2-generator hyperbolic 3-manifolds with a single cusp (original) (raw)

Genus 2 Closed Hyperbolic 3-MANIFOLDS of Arbitrarily Large Volume

arXiv: Geometric Topology, 2002

We describe a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume. The purpose of this note is to advertise the existence of a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume. The class described here consists merely of appropriate Dehn fillings on 2-bridge knots. That this class has the properties claimed follows directly from (4), (6), and the Gromov-Thurston 2…-Theorem. The existence of such a class of hyperbolic 3-manifolds is known, as pointed out by Cooper (3), who mentions that branched covers of the figure 8 knot provide another such class. We believe that the existence of such a class deserves to be more widely known. For general definitions and properties concerning knot theory, see (7) or (8). The following definition and theorem are due to M. Lackenby. Definition 1. Given a link diagram D, we call a complementary region having two crossings in its boundary a bigon region. A twist is a sequence v1;:::;vl of vertices such t...

Hyperbolic 3-manifolds with two generators

Communications in Analysis and Geometry, 1996

We show that if there are two parabolic elements that generate a non-elementary Kleinian group that is not free, then there is a universal upper bound of two on the "length" of each of those parabolics, length being measured in a canonical choice of cusp boundaries. Moreover, there is a universal upper bound of ln(4) on the "distance" between those parabolics, where the distance between them is the distance between a pair of horoballs corresponding to the canonical cusps. We prove a variety of results with these, the most interesting of which is: An orientable finite volume hyperbolic 3-manifold that has fundamental group generated by two parabolic isometries must be a 2-bridge knot or link complement.

Unknotting Tunnels, Bracelets and the Elder Sibling Property for Hyperbolic 3-MANIFOLDS

Journal of the Australian Mathematical Society, 2013

An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic alpha\alpha alpha in a hyperbolic 3-manifold MMM, we find sufficient conditions for it to be an unknotting tunnel. In particular, if alpha\alpha alpha corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic alpha\alpha alpha that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at infty\infty infty is connected to a larger horoball by a lift of alpha\alpha alpha. Such an alpha\alpha alpha with length less than ln(2)\ln (2)ln(2) is then shown to be an unknotting tunnel.

Fuchsian and quasi-Fuchsian surfaces in hyperbolic 3 – manifolds

2008

Given a noncompact quasi-Fuchsian surface in a finite volume hyperbolic 3–manifold, we introduce a new invariant called the cusp thickness, that measures how far the surface is from being totally geodesic. We relate this new invariant to the width of a surface, which allows us to extend and generalize results known for totally geodesic surfaces. We also show that checkerboard surfaces provide examples of such surfaces in alternating knot complements and give examples of how the bounds apply to particular classes of knots. We then utilize the results to generate closed immersed essential surfaces.

Quasi-Fuchsian surfaces in hyperbolic knot complements

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1993

Examples of hyperbolic knots in S3 are given such that their complements contain quasi-Fuchsian non-Fuchsian surfaces. In particular, this proves that there are hyperbolic knots that are not toroidally alternating.

Knots with only two strict essential surfaces

Proceedings of the Casson Fest, 2004

We consider irreducible 3-manifolds M that arise as knot complements in closed 3-manifolds and that contain at most two connected strict essential surfaces. The results in the paper relate the boundary slopes of the two surfaces to their genera and numbers of boundary components. Explicit quantitative relationships, with interesting asymptotic properties, are obtained in the case that M is a knot complement in a closed manifold with cyclic fundamental group.

Closed Incompressible Surfaces in Knot Complements

2000

In this paper we show that given a knot or link K in a 2n-plat projection with n 3a nd m5, where m is the length of the plat, if the twist coecients ai;j all satisfy jai;jj > 1t hen S 3 N ( K) has at least 2n 4 nonisotopic essential meridional planar surfaces. In particular if K is

Unknotting Tunnels of Hyperbolic Tunnel Number n Manifolds

2012

Unknotting Tunnels of Hyperbolic Tunnel Number n Manifolds Stephan D. Burton Department of Mathematics, BYU Master of Science Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n − 2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n − 1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the generalization of Adams’s conjecture to tunnel number n ≥ 2 manifolds may be false.