# A note on unavoidable sets for a spherical curve of reductivity four

## TL;DR

This paper investigates the properties of spherical curves with reductivity four, identifying unavoidable configurations and constructing examples of reduced spherical curves lacking certain polygon types.

## Contribution

It provides an unavoidable set of configurations for spherical curves with reductivity four and constructs a reduced spherical curve without 2-gons or specific 3-gons.

## Key findings

- Unavoidable set of configurations for reductivity four spherical curves.
- Existence of reduced spherical curves without 2-gons and certain 3-gons.
- Construction of explicit examples of such curves.

## Abstract

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configurations for a spherical curve with reductivity four is given by focusing on 5-gons. It has also been unknown if there exists a reduced spherical curve which has no 2-gons and 3-gons of type A, B and C. This paper gives the answer to the question by constructing such a spherical curve.