Simplifying branched covering surface-knots by an addition of 1-handles with chart loops

TL;DR

This paper introduces a method for simplifying branched covering surface-knots using 1-handles with chart loops, facilitating easier analysis and classification of these complex topological structures.

Contribution

It presents a novel simplification technique for branched covering surface-knots via 1-handles with chart loops, expanding the toolkit for topological surface-knot analysis.

Findings

Simplification reduces complex charts to union of free edges and 1-handles with loops.

Properties of the simplification process are systematically investigated.

The method aids in understanding the structure of branched covering surface-knots.

Abstract

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications.