scholarly journals INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

2006 ◽  
Vol 15 (07) ◽  
pp. 935-948 ◽  
Author(s):  
MARIO EUDAVE-MUÑOZ
Keyword(s):  
Lens Space ◽  
Regular Neighborhood ◽  
Incompressible Surface ◽  
Special Position ◽  
Incompressible Surfaces ◽  
Toroidal Graph ◽  
Tunnel Number ◽  
Tunnel Number One Knots

Let M be S3, S1 × S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.

scholarly journals Invariant incompressible surfaces in reducible 3-manifolds

2018 ◽  
Vol 39 (11) ◽  
pp. 3136-3143 ◽  
Author(s):  
CHRISTOFOROS NEOFYTIDIS ◽  
SHICHENG WANG
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Incompressible Surface ◽  
Mapping Torus ◽  
Incompressible Surfaces ◽  
Hyperbolic Automorphism

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$ . As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$ , or the mapping torus of a hyperbolic automorphism.

scholarly journals CHARACTER VARIETIES OF ONCE-PUNCTURED TORUS BUNDLES WITH TUNNEL NUMBER ONE

2013 ◽  
Vol 24 (06) ◽  
pp. 1350048 ◽  
Author(s):  
KENNETH L. BAKER ◽  
KATHLEEN L. PETERSEN
Keyword(s):  
Boundary Component ◽  
Lens Space ◽  
Algebraic Sets ◽  
Character Varieties ◽  
Torus Bundles ◽  
Whitehead Link ◽  
Birational Equivalence ◽  
Punctured Torus ◽  
Tunnel Number ◽  
Trace Fields

We determine the PSL2(ℂ) and SL2(ℂ) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine "natural" models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally, we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL2(ℂ).

scholarly journals Identifying tunnel number one knots

1996 ◽  
Vol 48 (4) ◽  
pp. 667-688 ◽  
Author(s):  
Kanji MORIMOTO ◽  
Makoto SAKUMA ◽  
Yoshiyuki YOKOTA
Keyword(s):  
Tunnel Number ◽  
Tunnel Number One Knots

Arcbodies

1983 ◽  
Vol 94 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Maria Teresa Lozano
Keyword(s):  
Recent Work ◽  
Incompressible Surface ◽  
Incompressible Surfaces ◽  
Positive Genus

A Haken manifold is a compact, orientable, irreducible 3-manifold which contains a properly embedded 2-sided, incompressible surface of positive genus. These manifolds are important in connection with the work of Haken, Waldhausen and the more recent work of Thurston (8). Thus it is interesting to investigate criteria for testing incompressible surfaces on 3-manifolds.

scholarly journals Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries

2020 ◽  
Vol 29 (11) ◽  
pp. 2050075
Author(s):  
Sungmo Kang
Keyword(s):  
Dehn Surgery ◽  
Surface Slope ◽  
Addition Formula ◽  
Closed Curves ◽  
Tunnel Number ◽  
Genus Two ◽  
Surgery Formula ◽  
Tunnel Number One Knots

Suppose [Formula: see text] and [Formula: see text] are disjoint simple closed curves in the boundary of a genus two handlebody [Formula: see text] such that [Formula: see text] (i.e. a 2-handle addition along [Formula: see text]) embeds in [Formula: see text] as the exterior of a hyperbolic knot [Formula: see text] (thus, [Formula: see text] is a tunnel-number-one knot), and [Formula: see text] is Seifert in [Formula: see text] (i.e. a 2-handle addition [Formula: see text] is a Seifert-fibered space) and not the meridian of [Formula: see text]. Then for a slope [Formula: see text] of [Formula: see text] represented by [Formula: see text], [Formula: see text]-Dehn surgery [Formula: see text] is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435–472.]. In this paper, we show that there exists a meridional curve [Formula: see text] of [Formula: see text] (or [Formula: see text]) in [Formula: see text] such that [Formula: see text] intersects [Formula: see text] transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery [Formula: see text] can arise from a primitive/Seifert position of [Formula: see text] with [Formula: see text] its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in [Formula: see text] is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.

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