• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.1
• /
• pp.1-30
• /
• 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
• /
• v.60 no.1
• /
• pp.203-223
• /
• 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.

• Honam Mathematical Journal
• /
• v.40 no.3
• /
• pp.591-600
• /
• 2018

In [4] Miyazaki and Motegi constructed one family of knots in $S^3$ which admits Dehn surgery producing a Seifert-fibered space over $S^2$ with four exceptional fibers. On the other hand, in [3] using doubly hyper Seifert twisted torus knots, the author constructed six families of knots in $S^3$ which admit Dehn surgery yielding a Seifert-fibered space over $S^2$ with four exceptional fibers. It is questioned in [3] whether or not the family of the knots constructed in [4] belongs to one of the six families of the knots in [3]. In this paper, we give the positive answer for this question.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.313-321
• /
• 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
• /
• v.53 no.1
• /
• pp.273-301
• /
• 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
• /
• v.36 no.3
• /
• pp.623-639
• /
• 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
• /
• v.38 no.1
• /
• pp.49.1-49.1
• /
• 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.

• Kyungpook Mathematical Journal
• /
• v.58 no.1
• /
• pp.183-202
• /
• 2018

${\Delta}-moves$ are closely related with a Vassiliev invariant of degree 2. For classical knots, M. Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single ${\Delta}-move$. The first author extended the Okada's result for virtual knots by using a Vassiliev invariant of virtual knots of type 2 which is induced from the Kauffman polynomial of a virtual knot. The arrow polynomial is a generalization of the Kauffman polynomial. We will generalize this result by using Vassiliev invariants of type 2 induced from the arrow polynomial of a virtual knot and give a lower bound for the number of ${\Delta}-moves$ transforming $K_1$ to $K_2$ if two virtual knots $K_1$ and $K_2$ are related by a finite sequence of ${\Delta}-moves$.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.4
• /
• pp.349-356
• /
• 2006

The aim of this paper is to endow a monoid structure on the set S of all oriented knots(links) under the operation ${\biguplus}$, called addition of knots. Moreover, we prove that there exists a homomorphism of monoids between ($S_d,\;{\biguplus}$) to (N, +), where $S_d$ is a subset of S with an extra condition and N is the monoid of non negative integers under usual addition.