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.
A
In other words, it is unknown if there exists a spherical curve whose reductivity is four. An
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:
(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
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
In this section, Theorem 1.3 is shown.
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:
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
The 5-gons of type 2, 3 and 4 have the reflection symmetries
Now let a 5-gon be a part of a spherical curve on
A 5-gon of type 1 has two symmetries
A 5-gon of type 2 has the reflection symmetry
Two cyclic sequences represent the same 5-gon when they are transformed into each other by a single
Two cyclic sequences represent the same 5-gon when they are transformed into each other by a single
Thus, 5-gons are classified into the 56 types shown in Figure 21.
In this section, Theorem 1.4 is proved.
Let
Hence, the set
A
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.