• Title/Summary/Keyword: 2-bridge knot

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ON THE 2-BRIDGE KNOTS OF DUNWOODY (1, 1)-KNOTS

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.197-211
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    • 2011
  • Every (1, 1)-knot is represented by a 4-tuple of integers (a, b, c, r), where a > 0, b $\geq$ 0, c $\geq$ 0, d = 2a+b+c, $r\;{\in}\;\mathbb{Z}_d$, and it is well known that all 2-bridge knots and torus knots are (1, 1)-knots. In this paper, we describe some conditions for 4-tuples which determine 2-bridge knots and determine all 4-tuples representing any given 2-bridge knot.

SIMPLE LOOPS ON 2-BRIDGE SPHERES IN HECKOID ORBIFOLDS FOR THE TRIVIAL KNOT

  • Lee, Donghi;Sakuma, Makoto
    • East Asian mathematical journal
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    • v.32 no.5
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    • pp.717-728
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    • 2016
  • In this paper, we give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even Heckoid orbifold for the trivial knot to be null-homotopic, peripheral or torsion in the orbifold. We also give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for the trivial knot to be homotopic in the orbifold.

THE AMPHICHEIRAL 2-BRIDGE KNOTS WITH SYMMETRIC UNION PRESENTATIONS

  • Toshifumi Tanaka
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.421-431
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    • 2024
  • In this paper, we characterize amphicheiral 2-bridge knots with symmetric union presentations and show that there exist infinitely many amphicheiral 2-bridge knots with symmetric union presentations with two twist regions. We also show that there are no amphicheiral 3-stranded pretzel knots with symmetric union presentations.

The Tunnel Number One Knot with Bridge Number Three is a (1, 1)-knot

  • Kim, Soo Hwan
    • Kyungpook Mathematical Journal
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    • v.45 no.1
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    • pp.67-71
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    • 2005
  • We call K a (1, 1)-knot in M if M is a union of two solid tori $V_1\;and\;V_2$ glued along their boundary tori ${\partial}V_1\;and\;{\partial}V_2$ and if K intersects each solid torus $V_i$ in a trivial arc $t_i$ for i = 1 and 2. Note that every (1, 1)-knot is a tunnel number one knot. In this article, we determine when a tunnel number one knot is a (1, 1)-knot. In other words, we show that any tunnel number one knot with bridge number 3 is a (1, 1)-knot.

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KNOTS WITH ARBITRARILY HIGH DISTANCE BRIDGE DECOMPOSITIONS

  • Ichihara, Kazuhiro;Saito, Toshio
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1989-2000
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    • 2013
  • We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g, b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g, b) = (0, 1), (0, 2).

Monodromy Groups on Knot Surgery 4-manifolds

  • Yun, Ki-Heon
    • Kyungpook Mathematical Journal
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    • v.53 no.4
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    • pp.603-614
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    • 2013
  • In the article we show that nondieomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family {$E(2)_K{\mid}K$ is a bered 2-bridge knot of genus g in $S^3$} admits a marked Lefschetz fibration structure which has the same monodromy group.

On the Polynomial of the Dunwoody (1, 1)-knots

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • Kyungpook Mathematical Journal
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    • v.52 no.2
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    • pp.223-243
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    • 2012
  • There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in $\mathbb{S}^3$ is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.