### Article

KYUNGPOOK Math. J. 2019; 59(4): 821-834

**Published online** December 23, 2019

Copyright © Kyungpook Mathematical Journal.

### A Note on Unavoidable Sets for a Spherical Curve of Reductivity Four

Kenji Kashiwabara, Ayaka Shimizu∗

Department of General Systems Studies, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8902, Japan

e-mail : kashiwa@idea.c.u-tokyo.ac.jp

Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan

e-mail : shimizu@nat.gunma-ct.ac.jp

**Received**: October 11, 2018; **Revised**: March 11, 2019; **Accepted**: March 18, 2019

### Abstract

The reductivity of a spherical curve is the minimal number of times a particular local transformation called an inverse-half-twisted splice is required to obtain a reducible spherical curve from the initial 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 this question by constructing such a spherical curve.

**Keywords**: knot projection, reductivity, spherical curve, unavoidable set.

### 1. Introduction

A ^{2}, where self-intersections, called ^{2}, and two spherical curves which are transformed into each other by a reflection are assumed to be the same spherical curve. A spherical curve is ^{2} which intersects ^{−1}, at a crossing of a spherical curve ^{−1} is a different local transformation from the splice called a “smoothing” in knot theory. Results from [3] show that for every pair of two nontrivial reduced spherical curves ^{−1}s and its inverses which transform ^{−1}s and its inverses. The ^{−1}s, which are required to obtain a reducible spherical curve from

### Problem 1.1.([6])

In other words, it is unknown if there exists a spherical curve whose reductivity is four. An _{1}, shown in Figure 3, of configurations with outer connections for a spherical curve with reductivity four was given. The unavoidable set _{1} was obtained by the following two facts; the first one is that every nontrivial reduced spherical curve has a 2-gon or 3-gon [1]. The second one is that if a spherical curve has a 2-gon or a 3-gon of type A, B or C, then the reductivity is three or less [6]. The following problem was also posed in [6]:

### Problem 1.2.([6])

If the answer to Problem 1.2 is “yes”, then the answer to Problem 1.1 is also “yes”. However, the following theorem gives the negative answer to Problem 1.2:

### Theorem 1.3

(See Figures 7 and 8 in Section 2.) In [5], 4-gons were classified into 13 types as shown in Figure 4 and the unavoidable set _{2}, in Figure 3, for a spherical curve with reductivity four was given by combining 3-gons and 4-gons based on an unavoidable set _{1} in Figure 5 for a nontrivial reduced spherical curve which was obtained in [6] in the same way to the four-color-theorem. Note that a necessary condition for a spherical curve with reductivity four was also given using the notion of the warping degree in [4]. In this paper, 5-gons are classified in a systematic way which can be used for general

### Theorem 1.4

_{3}

Theorem 1.4 would be useful for constructing a spherical curve with reductivity four (or showing that there are no such spherical curves), or detecting the reductivity for spherical curves which have no 2-gons and 3-gons of type A, B and C. The rest of the paper is organized as follows: In Section 2, Theorem 1.3 is shown. In Section 3, 5-gons are classified into 56 types. In Section 4, Theorem 1.4 is proved. In

### 2. Proof of Theorem 1.3

In this section, Theorem 1.3 is shown.

### Proof of Theorem 1.3

The spherical curves depicted in Figure 7 are reduced, and have no 2-gons and 3-gons of type A, B and C. The point is that there are no 2-gons, and all the 3-gons are of type D.

Note that the spherical curves shown in Figure 7 have the reductivity one, not four, because an inverse-half-twisted splice at a crossing at the middle 4-gons with a star derives a reducible spherical curve. Another example is shown in Figure 8. The reductivity of the spherical curve in Figure 8 is not four because it has a 4-gon, with a star in the figure, of type 4a; it is shown in [5] that if a spherical curve has a 4-gon of type 4a, then the reductivity is three or less.

In [6], a reduced spherical curve which has no 2-gons and 3-gons of type A and B was given. Further spherical curves are shown in Figure 9.

### 3. 5-gons

In this section, 5-gons are classified with respect to the outer connections by a systematic way which can be used for 6-gons or more:

### Lemma 3.1

**Proof**

There are four types of 5-gons when relative orientations of the five sides are considered. The 5-gons of type 1 to 4 are illustrated in Figure 10, where one of the relative orientations are shown by arrows. Let _{1} defined by the following permutations

The 5-gons of type 2, 3 and 4 have the reflection symmetries _{2}, _{3} and _{4} defined by the following permutations, respectively:

Now let a 5-gon be a part of a spherical curve on ^{2}. Let

**Type 1**

A 5-gon of type 1 has two symmetries _{1}. Two cyclic sequences which can be transformed into each other by some _{1} and orientation reversing (denoted by _{1}(_{1} and

abcde, abced =abdce =acbde =acdeb =aebcd ,abdec =abecd =acdba =adbce =adebc ,abedc =adcbe =adecb =aecdb =aedbc ,acebd, acedb =acbed =adceb =aebdc =aecbd, adbec, aedcb .

**Type 2**

A 5-gon of type 2 has the reflection symmetry _{2}. Two cyclic sequences represent the same 5-gon when they are transformed into each other by a single _{2} and orientation reversing _{2} and

abcde, abced =aebcd, abdce =acdeb, abdec =acdbe, abecd ,abedc =aecdb, acbde, acbed =aecbd, acebd, acedb =aebdc ,adbce =adebc, adbec, adcbe =adecb, adceb, aedbc, aedcb .

**Type 3**

Two cyclic sequences represent the same 5-gon when they are transformed into each other by a single _{3} and orientation reversing _{3} and

abcde, abced, abdce =aebcd, abdec =adebc, abecd =adbce ,abedc =aedbc, acbde =acdeb, acbed =acedb, acdbe, acebd, adbec ,adcbe =aecdb, adceb =aecbd, adecb, aebdc, aedcb .

**Type 4**

Two cyclic sequences represent the same 5-gon when they are transformed into each other by a single _{4} and orientation reversing _{4} and

abcde, abced =abdce, abdec =abecd, abedc, acbde =aebcd ,acbed =aebdc, acdbe =adebc, acdeb, acebd, acedb =adceb ,adbce, adbec, adcbe =aedbc, adecb =aecdb, aecbd, aedcb .

Thus, 5-gons are classified into the 56 types shown in Figure 21.

### 4. Proof of Theorem 1.4

In this section, Theorem 1.4 is proved.

### Proof of Theorem 1.4

Let _{2} in Figure 5 is also an unavoidable set for _{2} has already been discussed in Theorem 1 in [5]. Hence just the fourth one needs to be discussed here. Since the 3-gon should be of type D because 3-gons of type A, B and C make reductivity three or less, the 5-gon should be of type 2 or 4 with respect to the relative orientations of the sides (see Figure 16). Let

Hence, the set _{3} is an unavoidable set for a spherical curve of reductivity four.

### Appendix: 5-gons on chord diagrams

A ^{2}. In Figure 22, all the 5-gons of a spherical curve on chord diagrams are listed.

### Acknowledgements

The authors are grateful to the members of COmbinatoric MAthematics SEMInar (COMA Semi) for helpful comments. They also thank the timely help given by Yuki Miyajima in discovering reduced spherical curves without 2-gons and 3-gons of type A and B.

### Figures

_{1}and

_{2}for a spherical curve of reductivity four. Broken curves represent outer connections.

_{1}and

_{2}for a reduced spherical curve.

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