Genus one fibered knots in lens spaces

Following the classification of genus one fibered knots in lens spaces by Baker, we determine hyperbolic genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3-manifolds with left-orderable fundamental groups.

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Four-dimensional manifolds constructed by lens space surgeries of distinct types

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500699 ◽

2017 ◽

Vol 26 (11) ◽

pp. 1750069

Author(s):

Motoo Tange ◽

Yuichi Yamada

Keyword(s):

Fiber Surface ◽

Lens Space ◽

The Other ◽

Lens Spaces ◽

Dehn Surgery ◽

Integral Coefficient ◽

Torus Knots ◽

Torus Knot ◽

Simply Connected ◽

Genus One

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors’ previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge’s knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers.

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THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005019 ◽

2006 ◽

Vol 15 (09) ◽

pp. 1119-1129 ◽

Cited By ~ 2

Author(s):

A. CATTABRIGA

Keyword(s):

Fundamental Group ◽

Alexander Polynomial ◽

Lens Spaces ◽

Cyclic Covering ◽

Genus One

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces.

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BERGE'S KNOTS IN THE FIBER SURFACES OF GENUS ONE, LENS SPACE AND FRAMED LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216505003774 ◽

2005 ◽

Vol 14 (02) ◽

pp. 177-188 ◽

Cited By ~ 9

Author(s):

YUICHI YAMADA

Keyword(s):

Knot Theory ◽

Three Dimensional ◽

Lens Space ◽

Lens Spaces ◽

Dimensional Sphere ◽

View Point ◽

Complex Curves ◽

Resolution Of Singularity ◽

Kirby Calculus ◽

Genus One

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

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Tunnel number one, genus-one fibered knots

Communications in Analysis and Geometry ◽

10.4310/cag.2009.v17.n1.a1 ◽

2009 ◽

Vol 17 (1) ◽

pp. 1-16 ◽

Cited By ~ 3

Author(s):

Kenneth L. Baker ◽

Jesse E. Johnson ◽

Elizabeth A. Klodginski

Keyword(s):

Fibered Knots ◽

Tunnel Number ◽

Genus One

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Lehmer's Question, Knots and Surface Dynamics

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000618 ◽

2007 ◽

Vol 143 (3) ◽

pp. 649-661 ◽

Cited By ~ 2

Author(s):

DANIEL S. SILVER ◽

SUSAN G. WILLIAMS

Keyword(s):

Free Group ◽

Growth Rates ◽

Lens Spaces ◽

Surface Dynamics ◽

Lefschetz Numbers ◽

Fibered Knots ◽

Surface Homeomorphisms

AbstractLehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spacesL(n, 1),n> 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

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