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

TL;DR

This paper studies minimal charts of type (5,2) in the context of surface braids and 4-space embedded surfaces, focusing on their properties and applications.

Contribution

It introduces the concept of charts of type (5,2) and investigates their properties, particularly minimal charts, advancing understanding of surface braids and 4-dimensional topology.

Findings

Characterization of minimal charts of type (5,2)

Relationships between white vertices and chart structure

Implications for surface braid representations

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. 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 $(5,2)$.