• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.515-520
• /
• 2019

We give new evidence that "complicated" Heegaard surfaces behave like incompressible surfaces. More precisely, suppose that a closed connected orientable 3-manifold M contains a closed connected incompressible surface F which separates M into two (connected) components $M_1$ and $M_2$. Let S be a Heegaard surface of M. Our result is that if the Hempel distance of S is at least four, then S is isotoped so that $S{\cap}M_i$ is incompressible for each i = 1, 2.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.1
• /
• pp.67-77
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.6
• /
• pp.1989-2000
• /
• 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
• /
• v.54 no.5
• /
• pp.1851-1857
• /
• 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
• /
• v.46 no.5
• /
• pp.1019-1029
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.3
• /
• pp.735-740
• /
• 2015

Tomova [8] gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova [8, Theorem 10.3] can be improved in the case when there are two different bridge spheres for a link in $S^3$.

• Honam Mathematical Journal
• /
• v.35 no.4
• /
• pp.775-791
• /
• 2013

The twisted torus knots and the primitive/Seifert knots both lie on a genus 2 Heegaard surface of $S^3$. In [5], J. Dean used the twisted torus knots to provide an abundance of examples of primitive/Seifert knots. Also he showed that not all twisted torus knots are primitive/Seifert knots. In this paper, we study the other inclusion. In other words, it shows that not all primitive/Seifert knots are twisted torus knot position. In fact, we give infinitely many primitive/Seifert knots that are not twisted torus knot position.