# Minimal charts of type (3,3)

@article{Nagase2016MinimalCO, title={Minimal charts of type (3,3)}, author={Teruo Nagase and Akiko Shima}, journal={arXiv: Geometric Topology}, year={2016} }

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

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

- Mathematics
- 2020

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 $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=2$,…

## References

SHOWING 1-10 OF 10 REFERENCES

The lower bound of the w-indices of non-ribbon surface-links

- Mathematics
- 2004

Surface braidsare defined by Rudolph [13], [14] and Viro [18], correspondin g to oriented surface-links. Viro [18] and Kamada [10] proved th e Alexander’s theorem and Markov’s theorem for the…

Properties of minimal charts and their applications I

- Mathematics
- 2007

We study surface braids using the charts with minimal complexity. We introduce several terminology to describe minimal charts and investigate properties of minimal charts. We shall show that in a…

Properties of minimal charts and their applications II

- Mathematics
- 2009

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…

Gambits in charts

- Mathematics
- 2015

In this paper, we develop a method to change the label of a ring in a chart by C-moves and a stabilization where a ring is a simple closed curve consisting of edges of the same label which does not…

Braid and knot theory in dimension four

- Mathematics
- 2002

Basic notions and notation Classical braids and links: Braids Braid automorphisms Classical links Braid presentation of links Deformation chain and Markov's theorem Surface knots and links: Surface…

SURFACES IN R4 OF BRAID INDEX THREE ARE RIBBON

- Mathematics
- 1992

Any closed oriented surface embedded in R4 is described as a closed 2-dimensional braid and its braid index is defined. We study 2-dimensional braids through a method to describe them using graphs on…

Knotted Surfaces and Their Diagrams

- Mathematics
- 1997

Diagrams of knotted surfaces Moving knotted surfaces Braid theory in dimension four Combinatorics of knotted surface diagrams The fundamental group and the Seifert algorithm Algebraic structures…