On Crossing Changes for Surface-Knots

TL;DR

This paper explores crossing change operations on surface-knot diagrams, demonstrating how certain moves preserve exchangeable double curves and introducing a new numerical invariant for surface-knots.

Contribution

It introduces conditions under which crossing changes preserve exchangeable double curves and defines a novel invariant for surface-knot sets.

Findings

Finite sequences of Roseman moves can preserve exchangeable double curves.

A new numerical invariant called $du$-exchangeable set is defined.

Conditions for crossing change operations to preserve surface-knot properties.

Abstract

In this paper, we discuss the crossing change operation along exchangeable double curves of a surface-knot diagram. We show that under certain condition, a finite sequence of Roseman moves preserves the property of those exchangeable double curves. As an application for this result, we also define a numerical invariant for a set of surface-knots called $du$-exchangeable set.