• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.1-30
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• 2020

In this paper, we classify all twisted torus knots which are middle/hyper doubly Seifert. By the definition of middle/hyper doubly Seifert knots, these knots admit Dehn surgery yielding either Seifert-fibered spaces or graph manifolds at a surface slope. We show that middle/hyper doubly Seifert twisted torus knots admit the latter, that is, non-Seifert-fibered graph manifolds whose decomposing pieces consist of two Seifert-fibered spaces over the disk with two exceptional fibers.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.313-321
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• 2015

In this paper, we construct infinite families of knots in $S^3$ which admit Dehn surgery producing a Seifert-fibered space over $S^2$ with four exceptional fibers. Also we show that these knots are turned out to be satellite knots, which supports the conjecture that no hyperbolic knot in $S^3$ admits a Seifert-fibered space over $S^2$ with four exceptional fibers as Dehn surgery.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.273-301
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• 2016

In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in $S^3$ admitting Dehn surgery yielding such manifolds as done in [5].

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.623-639
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• 2021

We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.

• The Bulletin of The Korean Astronomical Society
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• v.38 no.1
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• pp.49.1-49.1
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• 2013

We have been carrying out near-infrared (NIR) spectroscopy as well as [Fe II] narrow band imaging observations of Cassiopeia A supernova remnant (SNR). In this presentation, we describe the spectral classification of the iron knots around the SNR. From eight long-slit spectroscopic observations for the iron-bright shell, we identified a total of 61 iron knots making use of a clump-finding algorithm, and performed principal component analysis in an attempt to spectrally classify the iron knots. Three major components have emerged from the analysis; (1) Iron-rich, (2) Helium-rich, and (3) Sulfur-rich groups. The Helium-rich knots have low radial velocities (${\mid}v_r{\mid}$ < 100 km/s) and radiate strong He I and [Fe II] lines, that match well with Quasi-Stationary Flocculi (QSFs) of circumstellar medium, while the Sulfur-rich knots show strong lines of oxygen burning materials with large radial velocity up to +2000 km/s, which imply that they are supernova ejecta (i.e. Fast-Moving Knots). The Iron-rich knots have intermediate characteristics; large velocity with QSF-like spectra. We suggest that the Iron-rich knots are missing "pure" iron materials ejected from the inner most region of the progenitor and now encountering the reverse shock.

• 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.

• 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.

• 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.

• 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.