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Akiko Shima

Publications and source records attributed to Akiko Shima.

15 recordsLinked to original sources
arXiv2025

The linear minimal 4-chart with three crossings

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by $Cross(\Gamma)$ the set of all the crossings of $\Gamma$, and we denote by $\Gamma_m$ the union of all the edges of label $m$. For a 4-chart $\Gamma$, if each connected component of the set $(\Gamma_1\cup \Gamma_3)-Cross(\Gamma)$ is acyclic, then $\Gamma$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops.

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arXiv2024

Properties of minimal charts and their applications X: charts of type $(5,2)$

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(5,2)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=5$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we investigate a minimal chart of type $(5,2)$.

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arXiv2023

Distinguishing surface-links with 4-charts with 2 crossings and 8 black vertices

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [11], [12], we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with 2 crossings and 8 black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links.

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arXiv2022

Properties of minimal charts and their applications IX: charts of type $(4,3)$

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensonal braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(4,3)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=4$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=3$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(4,3)$.

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arXiv2021

Properties of minimal charts and their applications VIII: charts of type $(7)$

Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(7)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=7$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(7)$.

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arXiv2020

Properties of minimal charts and their applications VII: charts of type $(2,3,2)$

Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(2,3,2)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=2$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=3$, and $w(\Gamma_{m+2}\cap\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(2,3,2)$.

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arXiv2020

Properties of minimal charts and their applications VI: the graph $\Gamma_{m+1}$ in a chart $\Gamma$ of type $(m;2,3,2)$

Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(m;2,3,2)$ if $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=2$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=3$, and $w(\Gamma_{m+2}\cap\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that if there is a minimal chart $\Gamma$ of type $(m;2,3,2)$, then each of $\Gamma_{m+1}$ and $\Gamma_{m+2}$ contains one of three kinds of graphs. In the next paper, we shall prove that there is no minimal chart of type $(m;2,3,2)$.

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arXiv2019

Minimal charts of type (3,3)

Let $Γ$ be a chart. For each label $m$, we denote by $Γ_m$ the "subgraph" of $Γ$ consisting of all the edges of label $m$ and their vertices. Let $Γ$ be a minimal chart of type $(m;3,3)$. That is, a minimal chart $Γ$ has six white vertices, and both of $Γ_m\capΓ_{m+1}$ and $Γ_{m+1}\capΓ_{m+2}$ consist of three white vertices. Then $Γ$ is C-move equivalent to a minimal chart containing a "subchart" representing a 2-twist spun trefoil or its "reflection".

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arXiv2019

Properties of minimal charts and their applications V: charts of type $(3,2,2)$

Let $Γ$ be a chart, and we denote by $Γ_m$ the union of all the edges of label $m$. A chart $Γ$ is of type $(3,2,2)$ if there exists a label $m$ such that $w(Γ)=7$, $w(Γ_m\capΓ_{m+1})=3$, $w(Γ_{m+1}\capΓ_{m+2})=2$, and $w(Γ_{m+2}\capΓ_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$.

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arXiv2018

The structure of a minimal $n$-chart with two crossings II: Neighbourhoods of $Γ_1\cupΓ_{n-1}$

Given a 2-crossing minimal chart $Γ$, a minimal chart with two crossings, set $α=\min\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$, and $β=\max\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$. In this paper we study the structure of a neighbourhood of $Γ_α\cupΓ_β$, and propose a normal form for 2-crossing minimal $n$-charts, here $Γ_α$ and $Γ_β$ mean the union of all the edges of label $α$ and $β$ respectively.

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arXiv2018

The structure of a minimal $n$-chart with two crossings I: Complementary domains of $Γ_1\cupΓ_{n-1}$

This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $Γ$ we denote by $Γ_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $Γ$ with exactly two crossings, we can show that the two crossings are contained in $Γ_α\capΓ_β$ for some labels $α<β$. In this paper, we study the structure of a disk $D$ not containing any crossing but satisfying $Γ\cap \partial D\subsetΓ_{α+1}\cup Γ_{β-1}$.

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arXiv2017

Minimal Charts

In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the fundamental theorems. In the future paper, we give an numeration of the charts with two crossings.

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arXiv2016

Separating a chart

In this paper, we shall show a condition for that a chart is C-move equivalent to the product of two charts, the union of two charts $Γ^*$ and $Γ^{**}$ which are contained in disks $D^*$ and $D^{**}$ with $D^*\cap D^{**}=\emptyset$.

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arXiv2016

Properties of minimal charts and their applications IV: Loops

We investigate minimal charts with loops, a simple closed curve consisting of edges of label $m$ containing exactly one white vertex. We shall show that there does not exist any loop in a minimal chart with exactly seven white vertices in this paper.

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