• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.491-502
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• 2000

The disjoint union of mapping class groups g,1 gives us a braided monoidal category so that it gives rise to a double loop space structure. We show that there exists a natural twist in this category, so it gives us a ribbon category. We show that there exists a natural twist in this category, so it gives us a ribbon category. We explicitly express this structure by showing how the twist acts on the fundamental group of the surface Sg,l. We also make an explicit description of this structure in terms of the standard Dehn twists, as well as in terms of Wajnryb's Dehn twists. We show that the inverse of the twist g for the genus g equals the Dehn twist along the fixed boundary of the surface Sg,l.

• 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.61 no.3
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• pp.585-595
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• 2024

We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve graph to that on Teichmüller space, (c) giving a partial answer of how much power is needed for Dehn twists to generate right-angled Artin subgroup of the mapping class group.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.601-614
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• 2016

Let $N_{g,s}$ denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski [12] obtained an explicit finite presentation for the mapping class group $\mathcal{M}(N_{g,s})$ of the surface $N_{g,s}$, where $s{\in}\{0,1\}$ and g + s > 3. Following this work, we obtain a finite presentation for the subgroup $\mathcal{T}(N_{g,s})$ of $\mathcal{M}(N_{g,s})$ generated by Dehn twists.

• 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.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.1-28
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• 2024

In this article, we associate a contact structure to the conjugacy class of a periodic surface homeomorphism, encoded by a combinatorial tuple of integers called a marked data set. In particular, we prove that infinite families of these data sets give rise to Stein fillable contact structures with associated monodromies that do not factor into products to positive Dehn twists. In addition to the above, we give explicit constructions of symplectic fillings for rational open books analogous to Mori's construction for honest open books. We also prove a sufficient condition for the Stein fillability of rational open books analogous to the positivity of monodromy for honest open books due to Giroux and Loi-Piergallini.