• Title, Summary, Keyword: Heegaard splitting

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ON (DISK, ANNULUS) PAIRS OF HEEGAARD SPLITTINGS THAT INTERSECT IN ONE POINT

  • Lee, Jung-Hoon
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.1
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    • pp.99-105
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    • 2009
  • Let $M=H_1{\cup}_SH_2$ be a Heegaard splitting of a 3-manifold M, D be an essential disk in $H_1$ and A be an essential annulus in $H_2$. Suppose D and A intersect in one point. First, we show that a Heegaard splitting admitting such a (D, A) pair satisfies the disjoint curve property, yet there are infinitely many examples of strongly irreducible Heegaard splittings with such (D, A) pairs. In the second half, we obtain another Heegaard splitting $M=H'_1{\cup}_{S'}H'_2$ by removing the neighborhood of A from $H_2$ and attaching it to $H_1$, and show that $M=H'_1{\cup}_{S'}H'_2$ also has a (D, A) pair with $|D{\cap}A|=1$.

A LOWER BOUND FOR THE GENUS OF SELF-AMALGAMATION OF HEEGAARD SPLITTINGS

  • Li, Fengling;Lei, Fengchun
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.67-77
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    • 2011
  • Let M be a compact orientable closed 3-manifold, and F a non-separating incompressible closed surface in M. Let M' = M - ${\eta}(F)$, where ${\eta}(F)$ is an open regular neighborhood of F in M. In the paper, we give a lower bound of genus of self-amalgamation of minimal Heegaard splitting $V'\;{\cup}_{S'}\;W'$ of M' under some conditions on the distance of the Heegaard splitting.

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).

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.

DISJOINT PAIRS OF ANNULI AND DISKS FOR HEEGAARD SPLITTINGS

  • SAITO TOSHIO
    • Journal of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.773-793
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    • 2005
  • We consider interesting conditions, one of which will be called the disjoint $(A^2,\;D^2)-pair$ property, on genus $g{\geq}2$ Heegaard splittings of compact orient able 3-manifolds. Here a Heegaard splitting $(C_1,\;C_2;\;F)$ admits the disjoint $(A^2,\;D^2)-pair$ property if there are an essential annulus Ai normally embedded in $C_i$ and an essential disk $D_j\;in\;C_j((i,\;j)=(1,\;2)\;or\;(2,\;1))$ such that ${\partial}A_i$ is disjoint from ${\partial}D_j$. It is proved that all genus $g{\geq}2$ Heegaard splittings of toroidal manifolds and Seifert fibered spaces admit the disjoint $(A^2,\;D^2)-pair$ property.

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 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.