Properties of Minimal Charts and Their Applications III

Abstract

Charts are oriented labeled graphs in a disk which correspond to surface braids. C-moves are local modifications of charts in a disk, which induces an ambient isotopy between the closures of the corresponding two surface braids. A chart is minimal if its complexity is minimal among the charts which are modified from the chart by C-moves. We investigate a disk whose boundary consists of edges of the same label, called a $k$-angled disk, for a minimal chart. In this paper we investigate 2-angled disks and 3-angled disks containing at most one white vertex in their interiors for a minimal chart.