• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.1-15
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• 2017

Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots K(p, q), where p and q are integers. The result implies that the rank of the Khovanov cohomology of K(p, q) is an invariant of p + q. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.385-395
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• 2013

To analyze the enzyme reaction on DNA knots and links, we study tangle embedding and the number of reaction. By using the quantum SU(3) invariant of knots and links we get a necessary condition for a tangle to be embedded in a knot or link. Moreover we give a relationship between the number of reactions and the changes of the value of quantum SU(3) invariant for the corresponding knots and links in a processive recombination.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.421-431
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• 2024

In this paper, we characterize amphicheiral 2-bridge knots with symmetric union presentations and show that there exist infinitely many amphicheiral 2-bridge knots with symmetric union presentations with two twist regions. We also show that there are no amphicheiral 3-stranded pretzel knots with symmetric union presentations.

• East Asian mathematical journal
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• v.16 no.1
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• pp.61-69
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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.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.449-461
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• 2006

Generalizing twist moves of classical knots, we introduce $t(a_1,{\cdots},a_m)$-moves of virtual knots for an $m$-tuple ($a_1,{\cdots},a_m$) of nonzero integers. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots and Gauss diagram formulae giving combinatorial presentations of finite type invariants. By using the Gauss diagram formulae for the finite type invariants of degree 2, we give a necessary condition for a virtual long knot K to be transformed to a virtual long knot K' by a finite sequence of $t(a_1,{\cdots},a_m)$-moves for an $m$-tuple ($a_1,{\cdots},a_m$) of nonzero integers with the same sign.

• 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.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.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.967-976
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• 2021

We obtain the list of prime knots with arc index 12 up to 16 crossings and their minimal grid diagrams. This is a continuation of the works [5] and [8] in which Cromwell matrices were generated to obtain minimal grid diagrams of all prime knots up to arc index 11. We provide minimal grid diagrams of the prime alternating knots with arc index 12. They are the 10 crossing prime alternating knots. The full list of 19,513 prime knots of arc index 12 up to 16 crossings and their minimal grid diagrams can be found in the arXiv [6].

• Journal of the Korean Society of Costume
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• v.60 no.10
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• pp.1-15
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• 2010

A Korean knot is one of the ornamental elements that our ancestors used intimately in their daily lives, and the diverse forms and structural features of the Korean knot have sufficient creative and aesthetic value for it to be recognized as one of beautiful products that was relished by individuals of the times. Starting from two strands, Korean knots make unique forms as they are overlapped or plaited, crossing each other in many ways. The forms of Korean knots were given names such as "nabi maedeup"(butterfly knots) and "gukwa maedeup" (chrysanthemum knots), in reference to things in the surrounding environment that were perceived as being similar in their appearance. It is considered that with their unique structure, such Korean knots may provide a good motif for creative design. As well, it is believed that combining the traditional beauty of Korean knots with a contemporary sensibility will lead to the creation of truly forward-looking design. Against this backdrop, this study aims to inquire into and analyze the formative characteristics and aesthetics of Korean knots, with an eye to their use in future design. In addition, it aims to help to put such historical knotting practices into practical and functional use in the future, through a study of previous uses of historical knotting practices with a modern sensibility. It is thus expected that this work will contribute to the inheriting and development of traditional culture, and ultimately to enhancing the status of Korean design in the world.

• 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.