• 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.

• 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.

• 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 $|D{\cap}A|=1$.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.113-122
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• 2010

In [4], we gave a condition for a pair of unknotting tunnels of a non-trivial tunnel number one link to give a genus three Heegaard splitting of the link complement. In this paper we prove the corresponding result for tunnel number one knots.

• 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.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.549-559
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• 2005

Given a non-Haken, non Seifert fibred manifold we describe an algorithm that takes 2 (not necessarily distinct) Heegaard surfaces and produces a complex with certain useful properties (Properties 5.1). Our main tool is Rubinstein and Scharlemann's Cerf theoretic work ([5]).

• 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.