An integral region choice problem on knot projection

TL;DR

This paper introduces a new region choice problem for knot projections, extending previous unknotting operations, and proves that solutions exist for all knot projections.

Contribution

It proposes an integral extension of the region crossing change operation and demonstrates the existence of solutions for all knot projections.

Findings

Solution exists for all knot projections

Extends Shimizu's region crossing change operation

Provides a new approach to knot projection manipulation

Abstract

In this paper we propose {\it a region choice problem} for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.