Properties of minimal charts and their applications XI: no minimal charts with exactly seven white vertices

TL;DR

This paper proves that minimal charts with exactly seven white vertices do not exist, providing insights into the structure of surface braids and embedded surfaces in 4-space.

Contribution

It establishes a non-existence result for minimal charts with seven white vertices, advancing understanding of chart properties in surface braid theory.

Findings

No minimal chart with exactly seven white vertices exists

Supports classification of charts based on white vertices

Enhances understanding of surface embeddings in 4-space

Abstract

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. In this paper, we shall show that there is no minimal chart with exactly seven white vertices.