• 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.

• 대한수학회보
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• 제52권1호
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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.

• 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.

• 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.

• 대한수학회지
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• 제47권3호
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• pp.585-598
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• 2010

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

• 대한수학회지
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• 제51권4호
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• pp.817-836
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• 2014

We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form T(2, n)#T(2, m). Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.

• East Asian mathematical journal
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• 제35권3호
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• pp.285-288
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• 2019

Let $F_n$, $n{\in}{\mathbb{N}}$ be the n - th Fibonacci number, and let (p, q) be one of ordered pairs ($F_{n+2}$, $F_n$) or ($F_{n+1}$, $F_n$). Then we show that the multiplicative inverse of q mod p as well as that of p mod q are again Fibonacci numbers. For proof of our claim we make use of well-known Cassini, Catlan and dOcagne identities. As an application, we determine the number $N_{p,q}$ of nonzero term of a polynomial ${\Delta}_{p,q}(t)=\frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}$ through the Carlitz's formula.

• 영어영문학
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• 제56권6호
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• pp.1295-1310
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• 2010

In her "After Language Poetry: Innovation and Its Theoretical Discontents" in Contemporary Poetics (2007), Marjorie Perloff spotted Steve McCaffery's and Lyn Hejinian's points of reference and opacity/transparency in poetic language, and theorizes in her perspicacious insights that poetic language is not a window, to be seen through, a transparent glass pointing to something outside it, but a system of signs with its own semiological interconnectedness. Providing a critique and contextualizing Perloff's argument, the purpose of this paper is to introduce a topological model for poetry, language, and theory and further to elaborate the relation between the theory and the practice of language poetry in terms of "the revolution of language." Jacques Lacan's poetics of knowledge and of the topology of the mind, in particular, that of "extimacy," can articulate the way how language poetry problematizes the opposition between inside and outside in the substance of language itself. In fact, as signifiers always refer not to things, but to other signifiers, signifiers becomes unconscious, and can say more than they actually says. The original signifiers become unconscious through the process of repression which makes a structure of multiple and polyphonic signifying chains. Language poets use this polyphonic language of the Other at Freudian "Another Scene" and Lacan's "Other." When the reader participates the constructive meanings, the locus of the language writing transforms itself into that of the Other which becomes the open field of language. The language poet can even manage to put himself in the between-the-two, a strange place, the place of the dream and of the Unheimlichkeit (uncanny), and suture between "the outer skin of the interior" and "the inner skin of the exterior" of the impossible real of definite meaning. The objective goal of the evacuation of meaning is all the same the first aspect suggested by the aims of the experimentalism by the language poetry. The open linguistic fields of the language poetry, then, will be supplemented by The Freudian "unconscious" processes of dreams, free associations, slips of tongue, and symptoms which are composed of this polyphonic language. These fields can be properly excavated by the methods and topological mapping of the poetics of extimacy and of the klein bottle.