• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.203-223
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• 2023

A generalized torsion element is an obstruction for a group to admit a bi-ordering. Only a few classes of hyperbolic knots are known to admit such an element in their knot groups. Among twisted torus knots, the known ones have their extra twists on two adjacent strands of torus knots. In this paper, we give several new families of hyperbolic twisted torus knots whose knot groups have generalized torsion. They have extra twists on arbitrarily large numbers of adjacent strands of torus knots.

• East Asian mathematical journal
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• v.16 no.1
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• pp.71-79
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• 2000

We show that the intersection of two standard torus knots of type (${\lambda}_1$, ${\lambda}_2$) and (${\beta}_1$, ${\beta}_2$) induces an automorphism of the cyclic group ${\mathbb{Z}}_d$, where d is the intersection number of the two torus knots and give an elementary proof of the fact that all non-trivial torus knots are strongly invertiable knots. We also show that the intersection of two standard knots on the 3-torus $S^1{\times}S^1{\times}S^1$ induces an isomorphism of cyclic groups.

• The Bulletin of The Korean Astronomical Society
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• v.35 no.2
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• pp.68.1-68.1
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• 2010

Cassiopeia A supernova remnant is a young (~330 yr) remnant of Type IIb SN explosion with a massive progenitor. It shows two distinct optical knots; fast moving ejecta knots (FMKs) and quasi stationary circumstellar knots (QSFs). These knots offer an unique opportunity to explore the details of the explosion and also the end state evolution of the Type IIb SN progenitor. We have obtained NIR long-slit (30") spectra of 7 positions around the bright rim of Cas A in [Fe II] 1.644 micron using Triplespec which is a cross-dispersed near-infrared spectrograph that provides continuous wavelength coverage from 0.95-2.46um at intermediate resolution of 2700. Most of the FMKs show strong sulfur, silicon, and iron forbidden lines but no hydrogen or helium lines. The QSFs, on the other hand, show a much richer spectrum with strong hydrogen, helium, and iron lines, but no sulfur and silicon lines. We measure their fluxes and radial velocities, and derive their physical parameters such as electron density and temperature. We also measure the proper motion of these knots from two [Fe II] 1.644 micron images obtained at 3-year interval. We analyze the physical properties of these knots and discuss the evolution and explosion of the progenitor of Cas A.

• East Asian mathematical journal
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• v.38 no.1
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• pp.107-127
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• 2022

In this paper, we discuss the relationship between doubly-pointed Heegaard diagrams of (1, 1)-knots in lens spaces and Dunwoody 3-manifolds, and then give explicit representations of n-fold cyclic branched coverings of all (1, 1)-knots in S³ up to 10 crossings in Rolfsen's knot table as Dunwoody 3-manifolds.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.101-108
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• 2003

In this paper we study the connections between cyclic presentations of groups and cyclic branched coverings of (1, 1)- knots. In particular, we prove that every π-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus π.

• East Asian mathematical journal
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• v.36 no.5
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• pp.537-545
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• 2020

We investigate some polynomial invariants for virtual knots via virtualization moves. We define two types of polynomials W_G(t) and S²_G(t) from the definition of the index polynomial for virtual knots. And we show that they are invariants for virtual knots on the quotient ring Z[t^±1]/I where I is an ideal generated by t² - 1.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.169-180
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• 2015

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

• 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.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.639-653
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• 2014

In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials $V_K(t)$ of classical links to the f-polynomials $f_K(A)$ of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots. In this paper, we give a necessary condition for a virtual knot invariant to be of finite type by using $t(a_1,{\cdots},a_m)$-sequences of virtual knots. Then we show that the higher derivatives $f_K^{(n)}(a)$ of the f-polynomial $f_K(A)$ of a virtual knot K at any point a are not of finite type unless $n{\leq}1$ and a = 1.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.329-334
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• 2007

By the works of Levine [2] and Rolfsen [5], [6], it is known that a local move called a crossing-change is strongly related to the Alexander invariant. In this note, we will consider to what degree the relationship is strong. Let K be a knot, and $K^{\times}$ the set of knots obtained from a knot K by a single crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander invariants $\{MK\}_{K{\in}\mathcal{K}}$ for a set of knots $\mathcal{K}$. Our main result is the following: If both $K_1$ and $K_2$ are knots with unknotting number one, then $MK_1=MK_2$ implies $MK_1^{\times}=MK_2^{\times}$. On the other hand, there exists a pair of knots $K_1$ and $K_2$ such that $MK_1=MK_2$ and $MK_1^{\times}{\neq}MK_2^{\times}$. In other words, the Gordian complex is not homogeneous with respect to Alexander invariants.