• Title/Summary/Keyword: Heegaard diagram

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REPRESENTATIONS OF n-FOLD CYCLIC BRANCHED COVERINGS OF (1, 1)-KNOTS UP TO 10 CROSSINGS AS DUNWOODY MANIFOLDS

  • Kim, Geunyoung;Lee, Sang Youl
    • East Asian mathematical journal
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    • v.38 no.1
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    • pp.107-127
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    • 2022
  • In this paper, we discuss the relationship between doubly-pointed Heegaard diagrams of (1, 1)-knots in lens spaces and Dunwoody 3-manifolds, and then give explicit representations of n-fold cyclic branched coverings of all (1, 1)-knots in S3 up to 10 crossings in Rolfsen's knot table as Dunwoody 3-manifolds.

CYCLIC PRESENTATIONS OF GROUPS AND CYCLIC BRANCHED COVERINGS OF (1, 1)-KNOTS

  • Mulazzani, Michele
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.101-108
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    • 2003
  • In this paper we study the connections between cyclic presentations of groups and cyclic branched coverings of (1, 1)- knots. In particular, we prove that every π-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus π.

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.

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.

ON CERTAIN CLASSES OF LINKS AND 3-MANIFOLDS

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • Communications of the Korean Mathematical Society
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    • v.20 no.4
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    • pp.803-812
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    • 2005
  • We construct an infinite family of closed 3-manifolds M(2m+ 1, n, k) which are identification spaces of certain polyhedra P(2m+ 1, n, k), for integers $m\;\ge\;1,\;n\;\ge\;3,\;and\;k\;\ge\;2$. We prove that they are (n / d)- fold cyclic coverings of the 3-sphere branched over certain links $L_{(m,d)}$, where d = gcd(n, k), by handle decomposition of orbifolds. This generalizes the results in [3] and [2] as a particular case m = 2.

ON INFINITE CLASSES OF GENUS TWO 1-BRIDGE KNOTS

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • Communications of the Korean Mathematical Society
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    • v.19 no.3
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    • pp.531-544
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    • 2004
  • We study a family of 2-bridge knots with 2-tangles in the 3-sphere admitting a genus two 1-bridge splitting. We also observe a geometric relation between (g - 1, 1)-splitting and (g,0)- splitting for g = 2,3. Moreover we construct a family of closed orientable 3-manifolds which are n-fold cyclic coverings of the 3-sphere branched over those 2-bridge knots.

SOME HYPERBOLIC SPACE FORMS WITH FEW GENERATED FUNDAMENTAL GROUPS

  • Cavicchioli, Alberto;Molnar, Emil;Telloni, Agnese I.
    • Journal of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.425-444
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    • 2013
  • We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.