Simplifying branched covering surface-knots by chart moves involving black vertices

TL;DR

This paper studies methods to simplify branched covering surface-knots, especially those with branch points, using chart moves involving black vertices, to achieve a standardized form with free edges and 1-handles.

Contribution

It introduces a new approach to simplify branched covering surface-knots with branch points through chart moves involving black vertices.

Findings

Simplification to a form with free edges and 1-handles achieved.

Properties of such simplifications are characterized for knots with branch points.

Chart moves involving black vertices are effective in the simplification process.

Abstract

A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications for the case when branched covering surface-knots have a non-zero number of branch points, using chart moves involving black vertices.