Let

Starting in [

Let

Generalizing the configuration space which was already constructed, we give the following filtration of

For

Here

Let

(i) An equilateral and equiangular

Consider an equilateral polygon whose bond angles are the same except for the last two ones. It can be considered as a mathematical model of a ringed hydrocarbon molecule. The space

The space

The purpose of this paper is to study the topology of

This paper is organized as follows. In Section

We first recall the results about

There is a homeomorphism:

Note that

Let

In the definition of

Then, we obtain the following description of

Hereafter, we use description (

The following definition and lemma will be used in Section

Let

We define the function

Then, we define

We define the function

Then, we define

When

We define

The linkages

The linkages

We set

For

For

For

Figures

For

For

For

Since the proofs are similar, we only prove (i). We set

On the other hand, the coordinate of the endpoint of

It is easy to see that the equation

First, it is easy to see that there is a homeomorphism

Let

We have

Moreover, the element of

When

We have

When

Case (iv) does not occur for

Assume that

Then, the topological type of

When

When

When

Regarding the Main Theorem (iii) for

Regarding the Main Theorem (v) (b) for

We fix

We define the projection

Let

We write an element of

Then, the following assertions hold:

When

When

When

When

We construct the following commutative diagram:

First, we set

Here, we define the following:

If we define the map

Second, we define the map

Fourth, we define the maps

We claim that for

In fact, we can identify

Now since the proofs of Lemma 2(i), (ii), and (iii) are similar, we prove only (iii). Note that for each

Using this fact, it is easy to see that

Combining (

(iv) By combining (

We claim that when

From this, we compute

The following lemma will be used in Section

Let

The lemma is well known (see, for example, ([

The most difficult case for the proof of the Main Theorem is the item (iv) for

Let

We define the function

The function

Now since

Note that

Since the case for

It is easy to prove the item (see also [

From Lemma 2(i), we have

The item follows from Lemma 2(ii).

The case for

Since the arguments are easy, we prove only the item (b). From the continuity of the deformation of

In addition to

Next, we determine

About (

In order to emphasize P, we write

By Lemma 2(iv), there is a fiber bundle:

We show that (

Consider the space

Moreover, since the base space is contractible, (

We have determined the topological type of

On the other hand, for the case

No data were used to support this study.

The author declares that there are no conflicts of interest regarding the publication of this paper.

This study was supported by JSPS KAKENHI (grant no. 15K04877).