A spherical curve is a closed curve on S², where self-intersections, called crossings, are double points intersecting transversely. In this paper, spherical curves are considered up to ambient isotopy of S², 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 trivial if it has no crossings. A spherical curve P is reducible if one can draw a circle on S² which intersects P transversely at just one crossing of P. Otherwise, it is said to be reduced. An inverse-half-twisted splice, denoted by HS⁻¹, at a crossing of a spherical curve P is a splice on P which yields another spherical curve (not a link projection) as shown in Figure 1. An inverse-half-twisted splice does not preserve an orientation of a spherical curve. Hence HS⁻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 P and P′, there exists a finite sequence of HS⁻¹s and its inverses which transform P into P′ such that a spherical curve at each step of the sequence is also reduced. This implies that all nontrivial reduced spherical curves are connected by HS⁻¹s and its inverses. The reductivity of a nontrivial spherical curve P is defined to be the minimal number of inverse-half-twisted splices, HS⁻¹s, which are required to obtain a reducible spherical curve from P. The reductivity tells us how reduced a spherical curve is, like the connectivity in graph theory. In [6], it is shown that every nontrivial spherical curve has the reductivity four or less. Also, in [5] and [6], it is mentioned that there are infinitely many spherical curves with reductivity 0, 1, 2 and 3. We still don’t know the answer to the following question:

Problem 1.1.([6])

For any nontrivial spherical curve, is the reductivity three or less?

In other words, it is unknown if there exists a spherical curve whose reductivity is four. An unavoidable set of configurations for a spherical curve in a class is a set of configurations with the property that any spherical curve in the class has at least one member of the set (see, for example, [2]). It is important to find unavoidable sets for a spherical curve of reductivity four from various viewpoints. In [6], 3-gons were classified into four types considering outer connections as shown in Figure 2 and the unavoidable set U₁, shown in Figure 3, of configurations with outer connections for a spherical curve with reductivity four was given. The unavoidable set U₁ 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])

Is the set consisting of a 2-gon, 3-gons of type A, B and C an unavoidable set for a reduced spherical curve?

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

There exists a reduced spherical curve which has no 2-gons and 3-gons of type A, B and C.

(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 U₂, in Figure 3, for a spherical curve with reductivity four was given by combining 3-gons and 4-gons based on an unavoidable set T₁ 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 n-gons (in Section 3) and another unavoidable set for a spherical curve with reductivity four is given:

Theorem 1.4

The set U₃shown in Figure 6 is an unavoidable set for a spherical curve with reductivity four.

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 Appendix, the 5-gons on chord diagrams are listed.

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.

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

5-gons of a spherical curve are divided into the 56 types in Figure 21 with respect to outer connections.