scholarly journals A Menagerie of SU(2)-Cyclic 3-Manifolds

2021 ◽  
Author(s):  
Steven Sivek ◽  
Raphael Zentner
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Hyperbolic Manifolds ◽  
Cyclic Surgery ◽  
Dehn Fillings

Abstract We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image, respectively, among geometric 3-manifolds that are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds that do not admit degree-1 maps to any Seifert Fibered manifold other than $S^3$ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four $SU(2)$-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.

scholarly journals Cyclic surgery on satellite knots

1991 ◽  
Vol 33 (2) ◽  
pp. 125-128 ◽  
Author(s):  
Xingru Zhang
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Dehn Surgery ◽  
Torus Knots ◽  
Torus Knot ◽  
Cyclic Surgery ◽  
Satellite Knots

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).

scholarly journals REDUCING DEHN FILLINGS AND SMALL SURFACES

2005 ◽  
Vol 92 (1) ◽  
pp. 203-223 ◽  
Author(s):  
SANGYOP LEE ◽  
SEUNGSANG OH ◽  
MASAKAZU TERAGAITO
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
The Other ◽  
Möbius Band ◽  
Dehn Fillings ◽  
Orientable Surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

scholarly journals An explicit relation between knot groups in lens spaces and those in S3

2018 ◽  
Vol 27 (08) ◽  
pp. 1850045
Author(s):  
Yuta Nozaki
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Lens Spaces ◽  
Alternative Proof ◽  
Cyclic Covering ◽  
Covering Map ◽  
Explicit Relation

For a cyclic covering map [Formula: see text] between two pairs of a 3-manifold and a knot each, we describe the fundamental group [Formula: see text] in terms of [Formula: see text]. As a consequence, we give an alternative proof for the fact that certain knots in [Formula: see text] cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group [Formula: see text] generated by the commutators and the [Formula: see text]th power of each element of [Formula: see text] plays a key role.

Klein slopes on hyperbolic 3-manifolds

2007 ◽  
Vol 143 (2) ◽  
pp. 419-447
Author(s):  
DANIEL MATIGNON ◽  
NABIL SAYARI
Keyword(s):  
Upper Bound ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
Dehn Fillings

AbstractThis paper is devoted to 3-manifolds which admit two distinct Dehn fillings producing a Klein bottle.LetMbe a compact, connected and orientable 3-manifold whose boundary contains a 2-torusT. IfMis hyperbolic then only finitely many Dehn fillings alongTyield non-hyperbolic manifolds. We consider the situation where two distinct slopes γ1, γ2produce a Klein bottle. We give an upper bound for the distance Δ(γ1, γ2), between γ1and γ2. We show that there are exactly four hyperbolic manifolds for which Δ(γ1, γ2) > 4.

scholarly journals A NOTE ON LENS SPACE SURGERIES: ORDERS OF FUNDAMENTAL GROUPS VERSUS SEIFERT GENERA

2011 ◽  
Vol 20 (04) ◽  
pp. 617-624 ◽  
Author(s):  
TOSHIO SAITO
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Dehn Surgery ◽  
Fundamental Groups

Let K be a non-trivial knot in the 3-sphere with a lens space surgery and L(p, q) a lens space obtained by a Dehn surgery on K. We study a relationship between the order p of the fundamental group of L(p, q) and the Seifert genus g of K. Considering certain infinite families of knots with lens space surgeries, the following estimation is suggested as a conjecture: [Formula: see text] except for (g, p) = (5, 19).

scholarly journals EXAMPLES OF REDUCIBLE AND FINITE DEHN FILLINGS

2010 ◽  
Vol 19 (05) ◽  
pp. 677-694 ◽  
Author(s):  
SUNGMO KANG
Keyword(s):  
Hyperbolic Manifolds ◽  
Dehn Filling ◽  
Dehn Fillings

If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperbolic manifolds admitting a reducible Dehn filling and a finite Dehn filling of every type: cyclic, dihedral, tetrahedral, octahedral and icosahedral.

scholarly journals Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

2021 ◽  
Author(s):  
Werner Müller
Keyword(s):  
Fundamental Group ◽  
General Result ◽  
Hyperbolic Manifold ◽  
Zeta Functions ◽  
Hyperbolic Manifolds ◽  
Reidemeister Torsion ◽  
Finite Dimensional ◽  
Orthogonal Representations ◽  
Ruelle Zeta Function ◽  
Complex Valued

AbstractThis paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $$s=0$$ s = 0 and its value at $$s=0$$ s = 0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

scholarly journals Integrality of volumes of representations

2021 ◽  
Author(s):  
Michelle Bucher ◽  
Marc Burger ◽  
Alessandra Iozzi
Keyword(s):  
Finite Volume ◽  
Hyperbolic Manifolds ◽  
3 Dimensional ◽  
Definition Of ◽  
Dehn Fillings

AbstractLet M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation $$\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)$$ ρ : π 1 ( M ) → Isom + ( H n ) , properly normalized, takes integer values if n is even and $$\ge 4$$ ≥ 4 . If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

THE SIMPLEST HYPERBOLIC KNOTS

1999 ◽  
Vol 08 (03) ◽  
pp. 279-297 ◽  
Author(s):  
PATRICK J. CALLAHAN ◽  
JOHN C. DEAN ◽  
JEFFREY R. WEEKS
Keyword(s):  
Ad Hoc ◽  
Crossing Number ◽  
Hyperbolic Manifolds ◽  
Torus Knots ◽  
Initial Observation ◽  
Knot Complements ◽  
Standard Notion ◽  
Dehn Fillings

While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which hyperbolic manifolds with 6 or fewer ideal tetrahedra are knot complements, and explicitly describe the corresponding knots in the 3-sphere. Thus, these 72 knots are the simplest knots according to this notion of complexity. Many of these knots have the structure of twisted torus knots. The initial observation that led to the project was the abundance of knot complements with small Seifert-fibered Dehn fillings among the census manifolds. Since many of these knots have rather large crossing number they do not appear in the knot tables. Our methods, while ad hoc, yield some detailed information about the knot complements as well as the manifolds that arise from exceptional surgeries on these knots.

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