• 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 $|D{\cap}A|=1$.

• 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.

• 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.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).

• 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.

• 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.

• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.699-715
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• 2018

In this paper, we show a proof of Gordon's Conjecture by using Qiu's labels and two new labels.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.1019-1029
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• 2009

We show that every bridge surface of certain types of (1, 1) prime knot has the disjoint curve property. Also we determine when a bridge surface of a pretzel knot of type (.2, 3, n) has the disjoint curve property.