• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.191-203
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• 2015

We introduce a quasitoric virtual braid and show that every virtual link can be obtained by the closure of a quasitoric virtual braid. Also, we show that the set of quasitoric virtual braids with n strands forms a group which is a subgroup of the n-virtual braid group.

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.179-193
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• 2003

The braid-permutation group is a group of welded braids which is the extension of Artin's braid groups by the symmetric groups. It is also described as a subgroup of the automorphism group of a free group. We also show that the plus-construction of the classifying space of the infinite braid-permutation group has the following two types of splittings BBP(equation omitted) B∑(equation omitted) $\times$ X, BBP(equation omitted) B $^{＋}$$\times$ Y＝S$^1$$\times$Y, where X, Y are some spaces.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.673-693
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• 1997

In this work, the irreducible complex representations of degree 4 of $B_4$, the braid group on 4 strings, are classified. There are 4 families of representations: A two-parameter family of representations for which the image of $P_4$, the pure braid group on 4 strings, is abelian; two families of representations which are the composition of an irreducible representation of $B_3$, the braid group on 3 strings, with a certain special homomorphism $\pi : B_4 \longrightarrow B_3$; a family of representations which are the tensor product of 2 irreducible two-dimensional representations of $B_4$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.3
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• pp.555-561
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• 2010

For the pure braid groups, this paper gives a positive finite presentation whose generators are the standard Artin generators, and determines whether the submonoid generated by the Artin generators is a Garside monoid or not.

• Kyungpook Mathematical Journal
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• v.49 no.1
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• pp.7-14
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• 2009

In this short note, we consider a family of linear representations of the braid group and the fundamental group of a punctured torus bundle over the circle. We construct an irreducible (special) unitary representation of the fundamental group of a closed 3-manifold obtained by the Dehn filling.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.445-454
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• 2010

The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra $TL_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector space underlying the Temperley-Lieb algebra $TL_n$. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno's theorem, using the interpretation of simple elements as non-crossing partitions.

• Korean Journal of Mathematics
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• v.19 no.4
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• pp.409-421
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• 2011

We study a natural map from the braid group to the mapping class group which is called Harer map. It is rather new and different from the classical map which was studied in 1980's by F. Cohen, J. Harer et al. We show that this map is homologically trivial for most coefficients by using the fact that this map factors through the symmetric group.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.865-877
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• 2013

The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\mathcal{M}$, as the disjoint union of the braid groups $\mathcal{B}$ does. We give a concrete and geometric meaning of the braidings ${\beta}_{r,s}$ in $\mathcal{M}$. Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we can define an obvious map ${\phi}\;:\;B_g{\rightarrow}{\Gamma}_{g,1}$. We show that this map ${\phi}$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor ${\Phi}\;:\;\mathcal{B}{\rightarrow}\mathcal{M}$, the integral homology homomorphism induced by ${\phi}$ is trivial in the stable range.

• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.337-345
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• 2006

John Harer conjectured that the canonical map from braid group to mapping class group induces zero homology homomorphism. To prove the conjecture it suffices to show that this map preserves the first Araki-Kudo-Dyer-Lashof operation. To get information on this homology operation we need to investigate the image of braiding under the Harer map. The main result of this paper is to give both algebraic and geometric interpretations of the image of braiding under the Harer map. For this we need to calculate long chains of consecutive actions of Dehn twists on the fundamental group of surface.

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.247-251
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• 1988

This note gives a combinatorial treatment to the problem finding a generating set among conjugating automorphisms of a free group and to the method deciding when a conjugating endomorphism of a free group is an automorphism. Our group of pseudo-conjugating automorphisms can be thought of as a generalization of the artin's braid group.