• 호남수학학술지
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• 제37권2호
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• pp.235-244
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• 2015

It is known that there is an infinite family of general pretzel knots, each of which has Rasmussen s-invariant equal to the negative value of its signature invariant. For an instance, homologically ${\sigma}$-thin knots have this property. In contrast, we find an infinite family of 4-strand pretzel knots whose Rasmussen invariants are not equal to the negative values of signature invariants.

• East Asian mathematical journal
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• 제36권5호
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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.

• 대한수학회보
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• 제55권3호
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• pp.937-956
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• 2018

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm [26] to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

• 대한수학회논문집
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• 제33권2호
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• pp.591-604
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• 2018

We consider a family of ribbon 2-knots with trivial Alexander polynomial. We give nonabelian SL(2, C)-representations from the groups of these knots, and then calculate the twisted Alexander polynomials associated to these representations, which allows us to classify this family of knots.

• East Asian mathematical journal
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• 제32권1호
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• pp.77-84
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• 2016

We explicitly derive diagrams representing (1,1)-decompositions of rational pretzel knots $K_{\beta}=M((-2,\;1),\;(3,\;1),\;({\mid}6{\beta}+1{\mid},{\beta}))$ from four unknotting tunnels for ${\beta}=1,\;-2$ and 2.

• Kyungpook Mathematical Journal
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• 제48권2호
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• pp.255-280
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• 2008

We derive a formula for the colored Jones polynomial of 2-bridge knots. For a twist knot, a more explicit formula is given and it leads to a relation between the degree of the colored Jones polynomial and the crossing number.

• East Asian mathematical journal
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• 제16권1호
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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.

• 대한수학회지
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• 제51권6호
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• pp.1221-1250
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• 2014

In this paper, we construct infinite families of knots in $S^3$ which admit Dehn surgery producing a graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. In particular, we show that for any natural numbers a, b, c, and d with $a{\geq}3$ and $b,c,d{\geq}2$, there are knots in $S^3$ admitting a graph manifold Dehn surgery consisting of two Seifert-fibered spaces over the disk with two exceptional fibers of indexes a, b, and c, d, respectively.

• Kyungpook Mathematical Journal
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• 제54권4호
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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.

• 한국생활과학회지
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• 제10권2호
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• pp.189-203
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• 2001

The pulpous of this study is to investigated the kind and feature of the knots used in four categories, "living, clothing, ceremony, and decoration", after analysing and examining 42 knot remains of the folklore museum of On yang. The method were by old books and various kind's related books, history of knots and kind of knots investigated and analyze relics were actual proof investigation of real things and supplementary materials were photos, drawings, and museum's explanation used. The results are as is following ; First, a knot is twisted by two threads, formed in the process of twisting, become in equal shape, and made in symmetry. Second, In knot's names, there are sangkang, karakge, ankyongchip, jamjary, kukhwa, bol, byongary, memi, kkondiki, kong, yonbong which are easily seen in nature. Third, a knot was used in as living things in the era of the three Kingdoms, as a decoration such as Buddhism goods in Koryo. In the era of Cho sun, Because of the development of knot's kind and the variety of service, it was the target of restriction as a luxury. Fourth, as a result of investigating museum's relics, a knot was smaller than the chief object because that is a decoration. Fifth, there are 8 kinds for living and clothing which are simple knots "doramaetup, karakgemaetup, and sangtchokmaetup". Sixth, the complex knots such as kukhwamaetup, byongarymaetup, and 3bolkamkae emaetup are used on pangchang, chokja, nambawi, and chobawi. Seventh, there are 10 kinds for ceremony, the simple knots are twisted and the complex knots such as kukhwamaetp, maehwamaetup, nabimaetup, and sasaekgupoki are used. Eighth, there are 14 kinds for decoration, the simple knots such as doraemaetup, kakagemaetup, santchokmaetup, and ankyongchipmaetup and the applied knots such as kukhwamaetup, byongarymaemaetup, changkumaetup, nabimaetup, and seokssima etup are used. There are 42 knot remains in the folklore museum of On yang. Of them, there are 33 kinds in the present, 17 used. that is because only several knots have been used. Finally, to forget the knot's way, we will set up a plan to keep with it.