On addition of 1-handles with chart loops to 2-dimensional braids

TL;DR

This paper studies how adding 1-handles with chart loops can simplify 2-dimensional braids over surface-knots, introducing moves that serve as unbraiding operations to analyze their structure.

Contribution

It introduces 1-handle moves with chart loops and demonstrates their role as unbraiding operations for 2-dimensional braids.

Findings

1-handle moves can simplify 2D braids

Addition of 1-handles with chart loops unbraids the structure

The method provides a systematic way to analyze braid complexity

Abstract

A 2-dimensional braid over an oriented surface-knot $F$ is presented by a graph called a chart on a surface diagram of $F$. We consider 2-dimensional braids obtained by an addition of 1-handles equipped with chart loops. We introduce moves of 1-handles with chart loops, called 1-handle moves, and we investigate how much we can simplify a 2-dimensional braid by using 1-handle moves. Further, we show that an addition of 1-handles with chart loops is an unbraiding operation.