scholarly journals Totally geodesic surfaces in hyperbolic 3-manifolds

1991 ◽  
Vol 34 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Alan W. Reid
Keyword(s):  
Totally Geodesic ◽  
Knot Complements ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Chern Simons

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350072
Author(s):  
PRADTHANA JAIPONG
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Incompressible Surface ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Dehn Fillings

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

A noncommutative generalization of Thurston’s gluing equations

2017 ◽  
Vol 28 (12) ◽  
pp. 1750089
Author(s):  
Xavier Morvan
Keyword(s):  
Gauge Theory ◽  
Knot Theory ◽  
American Mathematical Society ◽  
Advanced Mathematics ◽  
Noncommutative Ring ◽  
Regular Functions ◽  
Knot Complements ◽  
Figure Eight Knot ◽  
Chern Simons ◽  
Deformation Variety

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, Kashaev, [Formula: see text]-groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105–130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern–Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds.

scholarly journals Hyperbolic knot complements without closed embedded totally geodesic surfaces

2000 ◽  
Vol 68 (3) ◽  
pp. 379-386 ◽  
Author(s):  
Kazuhiro Ichihara ◽  
Makoto Ozawa
Keyword(s):  
Torus Knots ◽  
Totally Geodesic ◽  
Essential Surface ◽  
Knot Complements ◽  
Double Torus ◽  
Geodesic Surfaces

AbstractIt is conjectured that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. In this paper, we show that there are no such surfaces in the complements of hyperbolic 3-bridge knots and double torus knots. Some topological criteria for a closed essential surface failing to be totally geodesic are given. Roughly speaking, sufficiently ‘complicated’ surfaces cannot be totally geodesic.

scholarly journals Totally geodesic surfaces and homology

2006 ◽  
Vol 6 (3) ◽  
pp. 1413-1428 ◽  
Author(s):  
Jason DeBlois
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces

scholarly journals Non-simple geodesics in hyperbolic 3-manifolds

1994 ◽  
Vol 116 (2) ◽  
pp. 339-351
Author(s):  
Kerry N. Jones ◽  
Alan W. Reid
Keyword(s):  
Closed Geodesic ◽  
Closed Geodesics ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

The Generalized Cuspidal Cohomology Problem

2006 ◽  
Vol 58 (4) ◽  
pp. 673-690 ◽  
Author(s):  
Anneke Bart ◽  
Kevin P. Scannell
Keyword(s):  
Tangent Vector ◽  
Group Cohomology ◽  
Natural Generalization ◽  
Finite Range ◽  
Totally Geodesic ◽  
First Order ◽  
Link Complement ◽  
Cuspidal Cohomology ◽  
Zero Group ◽  
Geodesic Surfaces

AbstractLet Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\ℍ3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H1(Γ, ℍ41). Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

scholarly journals Compressing totally geodesic surfaces

2002 ◽  
Vol 118 (3) ◽  
pp. 309-328 ◽  
Author(s):  
Christopher J. Leininger
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces
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