• Title, Summary, Keyword: branched cyclic covering

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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.

THE KNOT $5_2$ AND CYCLICALLY PRESENTED GROUPS

  • Kim, Goan-Su;Kim, Yang-Kok;Vesnin, Andrei
    • Journal of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.961-980
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    • 1998
  • The cyclically presented groups which arise as fundamental groups of cyclic branched coverings of the knot $5_2$ are studied. The fundamental polyhedra for these groups are described. Moreover the cyclic covering manifolds are obtained in terms of Dehn surgery and as two-fold branched coverings of the 3-sphere.

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HEEGAARD SPLITTINGS OF BRANCHED CYCLIC COVERINGS OF CONNECTED SUMS OF LENS SPACES

  • Kozlovskaya, Tatyana
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1851-1857
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    • 2017
  • We study relations between two descriptions of closed orientable 3-manifolds: as branched coverings and as Heegaard splittings. An explicit relation is presented for a class of 3-manifolds which are branched cyclic coverings of connected sums of lens spaces, where the branching set is an axis of a hyperelliptic involution of a Heegaard surface.

ON HYPERBOLIC 3-MANIFOLDS WITH SYMMETRIC HEEGAARD SPLITTINGS

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1119-1137
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    • 2009
  • We construct a family of hyperbolic 3-manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3-balls and prove that all these manifolds are cyclic branched coverings of the 3-sphere over certain family of links with two components. These extend some results from [5] and [10] concerning with the branched coverings of the whitehead link.

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.

COMPLEXITY, HEEGAARD DIAGRAMS AND GENERALIZED DUNWOODY MANIFOLDS

  • Cattabriga, Alessia;Mulazzani, Michele;Vesnin, Andrei
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.585-598
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    • 2010
  • We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

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.

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.