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

TL;DR

This paper proves that minimal charts of type (4,3), which describe certain embedded surfaces in 4-space, do not exist, advancing the understanding of surface braid representations.

Contribution

It establishes the non-existence of minimal charts of type (4,3), providing new insights into the structure of charts representing embedded surfaces.

Findings

No minimal chart of type (4,3) exists.

The result constrains possible configurations of surface braids.

Advances the classification of charts for embedded surfaces in 4-space.

Abstract

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 $(4,3)$.