• Korean Journal of Mathematics
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• 제23권1호
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• pp.115-128
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• 2015

Plumbing surfaces of links were introduced to study the geometry of the complement of the links. A basket surface is one of these plumbing surfaces and it can be presented by two sequential presentations, the first sequence is the flat plumbing basket code found by Furihata, Hirasawa and Kobayashi and the second sequence presents the number of the full twists for each of annuli. The minimum number of plumbings to obtain a basket surface of a knot is defined to be the basket number of the given knot. In present article, we first find a classification theorem about the basket number of knots. We use these sequential presentations and the classification theorem to find the basket number of all prime knots whose crossing number is 7 or less except two knots $7_1$ and $7_5$.

• Kyungpook Mathematical Journal
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• 제52권2호
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• pp.223-243
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• 2012

There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in $\mathbb{S}^3$ is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.

• 호남수학학술지
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• 제32권1호
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• pp.157-165
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• 2010

Silver and Whitten proved that every knot in $S^3$ is invertibly concordant to a hyperbolic knot by a series of Nakanishi's construction. We prove that every knot in $S^3$ is invertibly concordant to a nonhyperbolic prime knot by a simple one step satellite construction.

• 대한수학회논문집
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• 제32권1호
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• pp.193-200
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• 2017

Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx-uy = 1, p = x + y and q = u + v. Using this property, we show that$${\sum\limits_{1{\leq}i{\leq}x,1{\leq}j{\leq}v}}\;{t^{(i-1)q+(j-1)p}\;-\;{\sum\limits_{1{\leq}k{\leq}y,1{\leq}l{\leq}u}}\;t^{1+(k-1)q+(l-1)p}$$ is the Alexander polynomial ${\Delta}_{p,q}(t)$ of a torus knot t(p, q). Hence the number $N_{p,q}$ of non-zero terms of ${\Delta}_{p,q}(t)$ is equal to vx + uy = 2vx - 1. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8); Let q be a positive integer> 1 and let k be a positive integer. Then we have $$\begin{array}{rccl}(1)&N_{kq}+1,q&=&2k(q-1)+1\\(2)&N_{kq}+q-1,q&=&2(k+1)(q-1)-1\\(3)&N_{kq}+2,q&=&{\frac{1}{2}}k(q^2-1)+q\\(4)&N_{kq}+q-2,q&=&{\frac{1}{2}}(k+1)(q^2-1)-q\end{array}$$ where we further assume q is odd in formula (3) and (4). Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q - 2, q) in [5] agree with $N_{kq+2,q}$ and $N_{kq+q-2,q}$ respectively.

• 정보과학회지
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• 제1권1호
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• pp.72-74
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• 1983

Traditionally, polynomials have been used to approximte functions with prescribed values at a number of points(called the knots) on a given interal on the real line. The method of splines recently developed is more flexible. It approximates a function in a piece-wise fashion, by means of a different polynomial in each subinterval. The cubic spline gas ets origins in beam theory. It possessed continuous first and second deriatives at the knots and is characterised by a minimum curvature property which es rdlated to the physical feature of minimum potential energy of the supported beam. Translated into mathematical terms, this means that between successive knots the approximation yields a third-order polynomial sith its first derivatives continuous at the knots. The minimum curvature property holds good for each subinterval as well as for the whole region of approximation This means that the integral of the square of the second derivative over the entire interval, and also over each subinterval, es to be minimized. Thus, the task of determining the spline lffers itself as a textbook problem in discrete computer programming, since the integral of ghe square of the second derivative can be obviously recognized as the criterion function whicg gas to be minimized. Starting with the initial value of the function and assuming an initial solpe of the curve, the minimum norm property of the curvature makes sequential decision of the slope at successive knots (points) feasible. It is the aim of this paper to derive the cubic spline by the methods of computer programming and show that the results which is computed the all the alues in each subinterval of the spline approximations.

• 호남수학학술지
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• 제30권1호
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• pp.165-170
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• 2008

We will examine the stick number of knots on the cubic lattice which is called the lattice stick number. The lattice stick numbers of knots $3_1$ and $4_1$ are known as 12 and 14, respectively. In this paper, we will show that only $3_1$ and $4_1$ have representations of irreducible non-trivial polygons, both numbers of whose sticks parallel to the y-axis and the z-axis are exactly four.

• 복식
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• 제60권10호
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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.

• East Asian mathematical journal
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• 제36권3호
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• pp.359-364
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• 2020

It is known that a 2-generator knot K has a cyclic Alexander module ℤ[t, t^―1]/(Δ(t)) where Δ(t) is the Alexander polynomial of K. In this paper we explicitly show how to reduce 2-generator Alexander modules to cyclic ones by using Chiswell, Glass and Wilsons presentations of 2-generator knot groups $$<\;x,\;y\;{\mid}\;(x^{{\alpha}_1})^{y^{{\gamma}_1}},\;{\cdots}\;,\;(x^{{\alpha}_k})^{y^{{\gamma}_k}}\;>$$ where a^b = bab^-1.

• 대한조선학회지
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• 제11권1호
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• pp.60-67
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• 1974

포항종합제철(浦項綜合製鐵)은 제품출하시(製品出荷時) 해송비(海送比)를 약(約) 80% 정도(程度)로 잡고 있으며, 현재(現在) 하루 평균(平均) 약(約) 3,000ton가량의 제품(製品)이 출하(出荷)되고 있다. 따라서 강재품(鋼材品) 수송(輸送)에 push-barge system을 적용(適用)했을 경우에 각(各) 주요예상소비지(主要豫想消費地)의 1일(日) 예상소비량(豫想消費量)을 장래의 시설확장(施設擴張)을 고려(考慮)하여 1,000ton/day에서 4,000ton/day까지 가정하였고 속도(速度)는 연안화물선(沿岸貨物船)의 예(例)를 따라 7knots에서 13knots까지에 걸쳐 조사(調査)하였다. 회항시(廻航時) 화물(貨物)은 일단 없는 것으로 가정하였으며 몇가지 생각될 수 있는 것은 schedule중(中)에서 "1 tugboats: 3 barges", "2 tugboats: 4 barges", "3 tugboats: 5 barges"의 3가지만을 대상(對象)으로 하였다. 조사(調査)한 결과(結果) 소비량(消費量) 1,000ton/day인 경우에 포항-울산(浦項-蔚山) : 1,000 DWT barge, 8 knots, 1 tugboat: 3 barges 포항-부산(浦項-釜山) : 1,550 DWT barge, 8 knots, 1 tugboat : 3 barges 포항-인천(浦項-仁川) : 2,690 DWT barge, 9 knots, 2 tugboats : 4 barges 등이 최적(最適)인 것으로 나타났다. 또 자항화물선(自航貨物船)과 비교(比較)하면 같은 크기의 자항화물선(自航貨物船) $2.5{\sim}3.0$척(隻)에 해당(該當)하는 효율(效率)을 가진다. 이 결과(結果)로 미루어 보건대 push-barge system은 앞으로 연구개발(硏究開發)의 충분(充分)한 가치(價値)가 있는 것으로 생각된다.

• 대한관절경학회지
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• 제8권1호
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• pp.37-44
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• 2004

목적: 잠김(Locking)이 가능한 이동 매듭의 (sliding knot) 최적 매듭 유지력 (knot-holdingcapacity KHC)을 가지기 위한 추가적인 반 매듭의 (additional half-hitches) 개수를 알아보고자 하였다. 대상 및 방법: 4가지의 관절경적 이동 매듭법 (Duncan매듭법, Field 매듭법, Giant매듭법, SMC매듭법)을 대상으로 매듭 유지력을 실험하였다. 각각의 매듭을 만들기 위해 처음의 이동 매듭과 추가적인 5개의 반 매듭으로 구성된 6개의 연속적인 매듭을 만들었다. 각각의 추가된 반 매듭은 서로 교차하며 매듭의 지대 (post)도 교차하여 매는 방식으로 (reversing half-hitches with alternate posts, RHAPS) 하였다. 각각의 연속적인 매듭을 구성하기위해 No.2 Ethibond봉합사를 이용하여 각각의 매듭 형태에 12개의 매듭을 만들었다. 매듭의 인장 및 유지력 실험은 servohydraulic material testing system(Instron 8511, MTS, Minneapolis, MN)으로 주기적 부하(cyclic loading)를 시켜, 임상적으로 실패라 규정한 3 mm의 전위가 생길 때까지의 부하 (load to clinical failure). 매듭이 완전히 실패했을 때의 부하 강도 (load to ultimate failure), 그리고 실패 형태 (mode of failure)를 측정하였다. 결과: 추가적인 반 매듭이 없는 최초의 이동 매듭 자체로는 대부분 주기적 부하에 의해 매듭의 실패를 보였다. 주기적 부하 검사에서 각각의 추가적인 반 매듭이 증가할수록 평균 전위 값은 감소하였다. SMC 매듭법과 Giant 매듭법은 하나의 추가적인 반 매듭 이후로 0.1 mm이하의 전위 값을 보였고 Field와 Duncan 매듭법은 3개의 추가 반매듭이 필요하였다. SMC 매듭법과 Duncan 매듭법은 80 N을 견디기 위해 단 하나의 추가 반 매듭이 필요하였고, Field와 Giant 매듭법은 2개의 추가 반 매듭이 필요하였다. 100 N이상의 부하를 견디기 위해서는 SMC 매듭법은 3개의 추가 반 매듭이 필요하였고, 다른 3가지의 매듭법은 4개의 추가 반 매듭이 필요하였다. 추가 반 매듭이 증가함에 따라 봉합사는 풀리는 것보다 끊어지는 양상을 보였다. Duncan매듭법은 5개의 추가 반 매듭 이후에도 풀림현상을 보였고, 다른 3개의 매듭법은 3개의 추가 반 매듭 이후엔 75%이상이 봉합사의 풀림 현상보다는 끊어지는 양상을 보였다. (SMC, Field 매듭법은 92%, Ciant 매듭법은 75%)결론: 어떤 매듭법이라도 이동 매듭법 만으로는 주기적 부하 검사를 견디지 못했다. 모든 종류의 실험에서 SMC 매듭법은 최소2개의 두개의 추가 반 매듭이 필요하였다. Duncan 매듭법은 최적 매듭 유지력을 위해 3개 이상의 추가 반 매듭이 필요하였다. 모든 매듭법에서 3개나 그 이상의 추가 반 매듭이 최적 유지력에서 정점에 가까운 양상을 보였다.