• 대한수학회보
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• 제35권2호
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• pp.235-258
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• 1998

We define a crossing of a link without referring to a specific projection of the link and describe a construction of a non-normalized Alexander polynomial associated to collections of such crossings of oriented links under an equivalence relation, called homology relation. The polynomial is computed from a special Seifert surface of the link. We prove that the polynomial is well-defined for the homology equivalence classes, investigate its relationship with the combinatorially defined Alexander polynomials and study some of its properties.

• Kyungpook Mathematical Journal
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• 제52권3호
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• pp.245-264
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• 2012

We determine lens surgeries (i.e. Dehn surgery yielding a lens space) along the n-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) n = 1 (i.e. the Whitehead link), and (2) one of surgery coefficients is 1, 2 or 3. Our interests are not only lens surgery itself but also how to apply the Alexander polynomial for this kind of problems.

• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.239-253
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• 2020

In this paper, we will find a Seifert matrix for a class of pretzel links with a certain symmetry. Using the symmetry, we find formulae for the Alexander polynomials, determinants and signatures of the pretzel links.

• 대한수학회지
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• 제49권5호
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• pp.993-1015
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• 2012

In this paper, we study the quantum sl($n$) representation category using the web space. Specially, we extend sl($n$) web space for $n{\geq}4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl($n$). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial ($n=0$) and Jones polynomial ($n=2$) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

• Kyungpook Mathematical Journal
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• 제51권3호
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• pp.261-281
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• 2011

In this paper, we will construct symmetric links by using the method adapted from the graph theory, and study a Seifert matrix of a symmetric link from the information of the Seifert matrix of the base link and the corresponding group action.