• Kyungpook Mathematical Journal
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• 제64권2호
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• pp.311-335
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• 2024

The fundamental quandle is a powerful invariant of knots, links and spatial graphs, but it is often difficult to determine whether two quandles are isomorphic. One approach is to look at quotients of the quandle, such as the n-quandle defined by Joyce [8]; in particular, Hoste and Shanahan [5] classified the knots and links with finite n-quandles. Mellor and Smith [12] introduced the N-quandle of a link as a generalization of Joyce's n-quandle, and proposed a classification of the links with finite N-quandles. We generalize the N-quandle to spatial graphs, and investigate which spatial graphs have finite N-quandles. We prove basic results about N-quandles for spatial graphs, and conjecture a classification of spatial graphs with finite N-quandles, extending the conjecture for links in [12]. We verify the conjecture in several cases, and also present a possible counterexample.

• Kyungpook Mathematical Journal
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• 제57권2호
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• pp.331-360
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• 2017

Unknotting numbers for torus knots and links are well known. In this paper, we present a new approach to determine the position of unknotting number crossing changes in a toric braid such that the closure of the resultant braid is equivalent to the trivial knot or link. Further we give unknotting numbers of more than 600 knots.

• 대한수학회논문집
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• 제38권3호
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• pp.913-924
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• 2023

We consider the number of colors for colorings of links by the symmetric group S₃ of degree 3. For knots, such a coloring corresponds to a Fox 3-coloring, and thus the number of colors must be 1 or 3. However, for links, there are colorings by S₃ with 4 or 5 colors. In this paper, we show that if a 2-bridge link admits a coloring by S₃ with 5 colors, then the link also admits such a coloring with only 4 colors.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제25권2호
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• pp.95-113
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• 2018

A mathematical knot is an embedded circle in ${\mathbb{R}}^3$. A fundamental problem in knot theory is classifying knots up to its numbers of crossing points. Knots are often distinguished by using a knot invariant, a quantity which is the same for equivalent knots. Knot polynomials are one of well known knot invariants. In 2006, J. Przytycki showed the effects of a n - move (a local change in a knot diagram) on several knot polynomials. In this paper, the authors review about knot polynomials, especially Jones polynomial, and give an alternative proof to a part of the Przytychi's result for the case n = 3 on the Jones polynomial.

• Kyungpook Mathematical Journal
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• 제64권3호
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• pp.511-518
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• 2024

A 2n-move is a local change for knots and links which changes 2n-half twists to 0-half twists or vice versa for a natural number n. In 1979, Yasutaka Nakanishi conjectured that the 4-move is an unknotting operation. This is still an open problem. It is known that the Γ-polynomial is an invariant for oriented links which is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that the 4k-move is not an unknotting operation for any integer k(≥ 2) by using the Γ-polynomial, and if Γ(K; -1) ≡ 9 (mod 16) then the knot K cannot be deformed into the unknot by a single 4-move. Moreover, we give a one-to-one correspondence between the value Γ(K; -1) (mod 16) and the pair (a₂(K), a₄(K)) (mod 2) of the second and fourth coefficients of the Alexander-Conway polynomial for a knot K.

• 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.

• 대한수학회지
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• 제38권2호
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• pp.437-468
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• 2001

This paper is an exposition of the relationship between Witten's functional integral and Vassiliev invariants.

• 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.

• Journal of applied mathematics & informatics
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• 제35권3_4호
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• pp.349-369
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• 2017

In last 30 years, mathematical tangle theory is applied to molecular biology, especially to DNA topology. The recent issues and research results of this topic are reviewed in this paper. We introduce a tangle which models an enzyme-DNA complex. The studies of 2-string tangle equations related to Topoisomerase II action and site-specific recombination is discussed. And 3-string tangle analysis of Mu-DNA complex, n-string tangle analysis ($n{\geq}4$) of DNA-enzyme synaptic complexes are also discussed.

• 대한수학회보
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• 제49권2호
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• pp.395-409
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• 2012

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let ${\pi}$ be a group and let M ${\rightarrow}$ N be a homomorphism between projective $\mathbb{Z}[{\pi}]$-modules such that $\mathbb{Z}_p\;{\otimes}_{\mathbb{Z}[{\pi}]}M{\rightarrow}\mathbb{Z}_p{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ is injective; for which other right $\mathbb{Z}[{\pi}]$-modules V is the induced map $V{\otimes}_{\mathbb{Z}[{\pi}]}\;M{\rightarrow}\;V{\otimes}_{\mathbb{Z}[{\pi}]}\;N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.