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

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.223-243
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• 2012

There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in $\mathbb{S}^3$ is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.155-158
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• 2004

We give a certain uniqueness properties for the fiber of the Hopf fibration on lens spaces.