Isolated regions of a link projection
Tumpa Mahato Ayaka Shimizu

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    24 April 2024
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A set of regions of a link projection is said to be isolated if any pair of regions in the set share no crossings. 
The isolate-region number of a link projection is the maximum value of the cardinality of an isolated set of regions of the link projection. 
In this paper, all the link projections of isolate-region number one are determined. 
Also, estimations for welded unknotting number and combinatorial way to find the isolate-region number are discussed, and a formula of the generating function of isolated-region sets is given for the standard projections of (2,n)-torus links.








§ INTRODUCTION


A knot is an embedding of a circle in S^3 and a link is an embedding of some circles in S^3. 
We assume a knot to be a link with one component. 
A link projection is a regular projection of a link on S^2 such that any intersection is a double point where two arcs intersect transversely. 
We call such an intersection a crossing. 
We call each connected part of S^2 divided by a link projection a region. 
It is well known that the number of regions is greater than the number of crossings by two[See, for example, <cit.>.] for each link projection.

Two regions x_a and x_b of a link projection L are said to be connected when they share a crossing on their boundaries. 
Otherwise, they are said to be disconnected. 
For example, the regions x_1 and x_2 are connected, whereas x_1 and x_8 are disconnected in Figure <ref>. 
A set of regions R_i= { x_a, x_b, … , x_c } is said to be isolated, or an isolated-region set, if any pair of regions x_α and x_β∈ R_i are disconnected. 
We assume that the empty set ∅ is isolated. 
The isolate-region number[The isolate-region number I(K) is similar but different from the “independent region number” IR(K) introduced in <cit.>; a crossing is fixed for IR(K).], I(L), of a link projection L is the maximum value of # (R_i) for all isolated-region sets R_i of L, where # (R_i) denotes the cardinality of R_i. 
For example, the isolate-region number of the knot projection K in Figure <ref> is three. 






We show the following theorem in Section <ref>.


A link projection L has isolate-region number one if and only if L is one of the link projections shown in Figure <ref>.






The concept of the isolate-region number was introduced in <cit.> to estimate the “warping degree” of knot projections (<cit.>). 
In Section <ref>, we investigate the warping degree and using the relation between the warping degree and the isolate-region number, we discuss the relation between the welded unknotting number u_w(D) of a knot diagram D and the isolate-region number I(D). 
We show an upper bound for u_w(D) (u_w(D) ≤ c(D) - I(D), Corollaries <ref>, <ref>) when D satisfies some conditions. 
Generally, however, determining the isolate-region number is not easy by looking only at the link projection of large number of crossings.  
In Section <ref>, a combinatorial method to find the isolate-region number using graphs is given. 
Moreover, from the graphs, we can see the distribution of the isolated-region sets by considering the generating function. 
We give a formula and recurrence relation of generating functions for the standard projections of (2,n)-torus links in Section <ref>.






The rest of the paper is organized as follows. 
In Section <ref>, the proof of Theorem <ref> is given. 
In Section <ref>, some properties of warping degree are discussed and applied to estimate the welded unknotting number of a knot diagram. 
In Section <ref>, two graphs, the region-connect graph and region-disconnect graph are introduced to find isolated-region sets. 
In Section <ref>, the generating function of isolated-region sets are discussed. 
In Section <ref>, the generating functions for the standard projections of (2,n)-torus links are discussed. 











§ PROOF OF THEOREM <REF>
 




In this section, we prove Theorem <ref>. 
For link projections M and N, a split sum and a connected sum are link projections obtained from M and N as shown in Figure <ref>. 
We say a link projection L is connected when L is not any split sum of link projections.
We say a link projection L is composite  when L is a connected sum of two nontrivial link projections. 
We have the following lemma. 






Every unconnected or composite link projection L has I(L) ≥ 2. 




When L is a split sum of M and N, take a region from M and N avoiding the region between them, and they are disconnected. 
When L is a connected sum of M and N, take a region from M-part and N-part avoiding the shared two regions. 
Thus, we have I(L) ≥ 2. 



A crossing of a link projection is said to be reducible if it has exactly three regions around it. 
A link projection L is reducible when L has a reducible crossing. 
We say L is irreducible when L is not reducible. 
Since a reducible link projection can be assumed to be a connected sum when it has at least two crossings, we have the following corollary. 


Every reducible link projection L with two or more crossings has I(L) ≥ 2. 



Next, we divide a link projection into two tangles by giving a circle C, where C intersects edges transversely. 
We assume each tangle T is on a disk D with ∂ D =C, and call each connected portion of D divided by T a region of T. 
In particular, if a region touches ∂ D, we call it an open region, and otherwise we call it a closed region. 
We remark that any pair of closed regions in opposite sides of C are disconnected. 
We show the following.



For any connected irreducible link projection with four or more crossings, we can draw a circle C so that the tangle on each side of C has a closed bigon or trigon. 





By the formula 

    2C_2 +C_3 = 8 + C_5 + 2C_6 + 3C_7 + …

for the number C_k of k-gons of a connected irreducible link projection shown in <cit.>, we have 2C_2 + C_3 ≥ 8. 


(0) When C_2 =0, i.e., C_3 ≥ 8, draw a circle C around a trigon as shown in Figure <ref>, (II). 
Then the tangle in the figure has a closed trigon. 
Since C_3 ≥ 8 >7 and  there are at most six regions around the trigon, the tangle on the other side has a closed trigon, too. 






(1) When C_2 =1, i.e., C_3 ≥ 6, draw a circle C around the bigon as shown in Figure <ref>, (I). 
Then, the tangle on the other side has closed trigons because C_3 ≥ 6 >4 and there are at most four regions around a bigon. 

(2) When C_2 =2, i.e., C_3 ≥ 4, draw C as shown in Figure <ref>, (I). 
Since C_2 + C_3 ≥ 6 >5, the tangle on the other side has a closed bigon or trigon, too. 

(3) When C_2 =3, i.e., C_3 ≥ 2, we consider the following cases. 

(3-1) When C_2 =3 and C_3 =2, choose any bigon and see (III) in Figure <ref>. We consider the further two cases whether a=c or not. 

(3-1-1) When a=c, we can draw a circle C as in Figure <ref>, (I) since we have C_2 + C_3 =5 >4 and the bigon has only three regions around it. 

(3-1-2) Suppose a ≠ c. Then neither b nor d is a bigon. 
Suppose a and c are connected and both bigons. 
Then the curve is closed with only three crossings. 
Hence this does not happen. 
When a and c are connected and either a or c is not a bigon, take C as shown in Figure <ref>, (I). 
Then the other side has a closed bigon, too.

(3-1-3) Suppose a ≠ c and they are disconnected bigons. 
Then both b and d are 4-gon or more. 
Take C as in Figure <ref>, (I). 
Then the other side of C has closed trigons.
Suppose a and c are disconnected and either a or c is not a bigon. 
Draw C as in Figure <ref>, (I). 
Then the other side has a closed bigon. 


(3-2) When C_2 =3 and C_3 ≥ 3, we can draw C as in Figure <ref>, (I), because C_2 + C_3 ≥ 6 >5. 

(4) When C_2=4, i.e., C_3 ≥ 0, choose any bigon and see (III) in Figure <ref>. 

(4-1) When a=c and it is a bigon, then the curve is closed with only two crossings. 
Hence this is not allowed to happen. 
When a=c and it is not a bigon, we can draw C as in Figure <ref>, (I) since inside C in the figure has at most three bigons. 

(4-2) When a ≠ c, we can draw C as shown in Figure <ref>, (I) since neither b nor d is a bigon, and inside C in the figure has at most three bigons, too. 

(5) When C_2 ≥ 5, choose any bigon and see (III) in Figure <ref>. 
If all of a, b, c, d are distinct bigons, then the curve is closed with only two crossings. Hence this does not happen. 
We can draw C as in Figure <ref>, (I) since inside C in the figure has at most four bigons. 



By Lemmas <ref>, <ref> and Corollary <ref>, we have the following.



Any link projection L with 4 or more crossings has I(L) ≥ 2. 




Now we prove Theorem <ref>. 



Proof of Theorem <ref>.
Since every unconnected projection has isolate-region number two or more by Lemma <ref>, it is sufficient to discuss connected projections. 
When the crossing number n=0, 1, there are just two link projections, the first and second knot projections in Figure <ref>, and they have isolate-region number one. 
When n=2,3, we have just two reduced link projections, the third and fourth projections in Figure <ref>, and they have isolate-region number one. 
By Corollaries <ref> and <ref>, there are no link projections of isolate-region number one for link projections with four or more crossings or reducible projections with two or more crossings. 



In addition to Corollary <ref>, we have the following proposition regarding the location of isolated regions. 


If an irreducible knot projection has an n-gon for n ≥ 4, there is a pair of disconnected regions around the n-gon. 



Take points a, b, c, d on edges of the n-gon and call the regions A, B, C, D, as shown in Figure <ref>. 
Since the knot projection is irreducible, we have A ≠ B, B ≠ C, C ≠ D. 
We also have A ≠ D since n ≥ 4. 

(1) If A ≠ C and A and C are disconnected, they are the regions. 

(2) If A=C, we can draw a simple curve α connecting a and c inside the region A=C. 
Since B and D cannot touch α, they are disconnected. 

(3) If A ≠ C and A and C are connected, we can draw a simple curve connecting a and c passing through A, C and one crossing p shared by A and C. 
Then, B and D are disconnected. 
Otherwise, B and D must share the crossing p to be connected, and we can draw a simple curve β connecting b and d which passing through B, D, and p. 
Then, at least one of the arcs between A and B loses the place to go since both α and β have only one crossing p on it. 





Also, we have the following corollary from Lemma <ref>. 


Any connected irreducible link projection L with 13 or more crossings has I(L) ≥ 3. 



By Lemma <ref>, L has a pair of disconnected regions of bigon or trigon, say r_1 and r_2. 
Then L has at most 12 regions which are connected to either r_1 or r_2. 
Since L has 15 or more regions, L has at least one region, say r_3, which is neither r_1 nor r_2 and is connected to neither r_1 nor r_2.






§ WARPING DEGREE, WELDED UNKNOTTING NUMBER AND ISOLATE-REGION NUMBER OF A KNOT DIAGRAM



In this section, we discuss the warping degree and welded unknotting number of a knot diagram and their relations to the isolate-region number. 
In Subsection <ref>, we discuss the relation between the warping degrees of a knot diagram and knot projection. 
In Subsection <ref>, we discuss the relation between the welded unknotting number, warping degree and isolate-region number of a knot diagram. 




 §.§ Warping degree



Let D be an oriented knot diagram with a base point b on an edge. 
A crossing point of D is said to be a warping crossing point of D_b if we encounter the crossing as an under crossing first when we travel D from b with the orientation. 
The warping degree of D_b, d(D_b), is the number of the warping crossing points of D_b. 
We can represent all the warping degrees by the “warping degree labeling” as shown in Figure <ref>.





The warping degree of D, d(D), is the minimal value of d(D_b) for all base points b of D (<cit.>). 
As shown in Figure <ref>, warping degree depends on the orientation; 
we have d(D)=1, whereas d(-D)=2, where -D is D with orientation reversed. 
In this paper, we define d(D)= min{ d(D), d(-D) }. 
For a knot diagram D, we denote by |D| the knot projection which is obtained from D by forgetting the crossing information. 
The warping degree of P, d(P), is the minimal value of d(D) for all oriented alternating diagrams D with |D|=P. 
Although each projection has four oriented alternating diagrams (see Figure <ref>), the following proposition claims it is sufficient to check only one alternating diagram with two orientations.


The equality d(D)=d(P) holds when D is an alternating diagram with |D|=P. 



As shown in <cit.>, we have d(-D_b)=c(D)-d(D_b) for any oriented knot diagram D with a base point b because the order of over crossing and under crossing is switched at every crossing. 
We also have d(D_b^*)= c(D)-d(D_b) for the mirror image D^* because the crossing information is changed at each crossing. 
Therefore, we have d(D_b^*)=d(-D_b), and d(D^*)=d(-D). 
Hence d(P) = min{ d(D),  d(-D),  d(D^*),  d(-D^*)  } = min{ d(D),  d(-D) } = d(D) for any alternating diagram D with |D|=P. 


A nonalternating knot diagram D is said to be almost alternating if D becomes alternating by a single crossing change. 
For almost alternating diagrams, we have the following proposition. 


Let D be an almost alternating diagram with |D|=P. 
Then d(D) ≤ d(P) holds. 
In particular, if an almost alternating diagram D has no monogons, d(D)=d(P)-1.



To prove Proposition <ref>, we prepare a further formula. 
Since we have d(-D)= min_b d(-D_b) = c(D) - max_b d(D_b) for any diagram D, we have

    d(D)= min{ d(D),  d(-D) } = min{min_b d(D_b),  c(D)- max_b d(D_b) }


for any knot diagram D. 
Hence, we can find the value of d(P) only from one oriented alternating diagram D with |D|=P by looking at the minimum and maximum values of warping degree labeling. 
We prove Proposition <ref>. 


Proof of Proposition <ref>. 
Let D be an almost alternating knot diagram which is obtained from an oriented alternating diagram D^A by a single crossing change at a crossing c. 
Let C_1 (resp. C_2) be a portion of D^A from a point just after the over crossing (resp. the under crossing) of c to a point just before the under crossings (resp. the over crossing) of c. 
Let C_1' (resp. C_2') be the corresponding portions of D to C_1 (resp. C_2). 
Each warping degree labeling on C_1' is smaller than the corresponding labeling on C_1 by one because c becomes a non-warping crossing point for base points on C_1' by the crossing change at c (<cit.>). 
In the same way, each labeling on C_2' is greater than C_2 by one.
Let d(D^A)=i. 
Then we have d(-D^A)=c(D)-i-1, and d(D^A)= min{ i,  c(D)-i-1 }. 

(1) When both C_1 and C_2 have crossings, their warping degree labeling are consisting of i and i+1. 
After the crossing change, the labels on C_1' are i-1,  i and the labels on C_2' are i+1,  i+2. 
Hence d(D)=i-1 =d(D^A)-1 and d(-D)= c(D)-i-2 = d(-D^A)-1, and therefore d(D)= d(D^A)-1. 

(2) When C_1 has no crossings and C_2 has crossings, 
the label of C_1 is only i+1 and the labels of C_2 are i+1,  i+2, and the labels of C_1' is i and that of C_2 are i+1,  i+2. 
Hence d(D)=i,  d(-D)=c(D)-i-2, and d(D) ≤d(D^A). 

(3) When C_1 has crossings and C_2 has no crossings, 
the labels of C_1 are i-1,  i and that of C_2 is only i, and the labeling of C_1' are i-1,  i and that of C_2' is i+1. 
Hence d(D)=i-1,  d(-D)= c(D) -i-1, and d(D) ≤d(D^A). 

(4) When neither C_1 nor C_2 has crossings, then c(D)=1, and D is not an almost alternating diagram. 

Therefore, we have d(D) ≤d(D^A)=d(P), where P=|D|. 
In particular, if D has no monogons, then both C_1 and C_2 must have crossings for any crossing c, and this case applies (1). 



Next, we show the following proposition. 


Let P be an irreducible knot projection with c(P) ≤  9. 
The inequality d(D) ≤ d(P) holds for any knot diagram D with |D|=P if and only if P is neither of the four projections illustrated in Figure <ref>.








To prove Proposition <ref>, we prepare Lemmas <ref>, <ref>, <ref>. 
All the irreducible knot projections with warping degree 1 and 2 are determined in <cit.>. 

<cit.>
Let P be an irreducible knot projection. 

(1) d(P)=1 if and only if P is one of the two projections depicted in Figure <ref>. 

(2) d(P)=2 if and only if P is one of the 16 projections depicted in Figure <ref>.








The following lemma gives an upper bounds for d(D) and d(P). 


The inequality d(D) ≤ (c(D)-1)/2 holds for any knot diagram D with c(D) ≥ 1. 
The inequality d(P) ≤ (c(P)-1)/2 holds for any knot projection P with c(P) ≥ 1.




By the inequality d(D)+d(-D) ≤ c(D)-1 shown in <cit.>, we have 2 d(D) ≤ c(D)-1, and therefore d(D) ≤ (c(D)-1)/2. 
Since this also holds for alternating diagrams D, we have d(P) ≤ (c(P)-1)/2. 



The span, s(D), of an oriented knot diagram D is defined to be the difference s(D)= max_b d(D_b) - min_b d(D_b).
By definition, we have s(D)=s(-D). 
Also, we have the following. 

<cit.>
The span s(D) is one if and only if D is an alternating diagram.



Using the above lemmas, we show Proposition <ref>. 



Proof of Proposition <ref>. 
We show that there exist no non-alternating diagrams D with |D|=P such that d(D) > d(P) under the condition. 


∙ When c(P)=3 or 4, we have d(P)=1 by Lemma <ref>, and there are no diagrams D with |D|=P satisfying d(D)>1 by Lemma <ref>. 

∙ When c(P)=5 or 6, we have d(P)=2 by Lemmas <ref>, <ref>, and there are no diagrams D satisfying d(D) >2 by Lemma <ref>. 

∙ When c(P)=7, d(P)=2 or 3 by Lemmas <ref>, <ref>. 
There are no diagrams D with |D|=P satisfying d(D) >3 by Lemma <ref>. 
When d(P)=2, each oriented alternating diagram D^A with |D^A|=P has warping degree labeling { 2, 3 } or { 4, 5 }. 
There are no nonalternating diagrams D with |D|=P such that the minimal label is greater than 2 and the maximal labeling is smaller than 5, because s(D)>1. 

∙ When c(P)=8, d(P)=2 or 3 by Lemmas <ref>, <ref>. 
When d(P)=2, P is one of the four knot projections depicted in Figure <ref> by Lemma <ref>. 
For each knot projection P, we have a diagram D with |D|=P such that d(D)=3 > 2=d(P) as shown in Figure <ref>. 



When d(P)=3, the warping degree labeling on D^A is { 3, 4 } or { 4, 5 }, and there are no nonalternating diagram D such that the minimal label is greater than 3 and the maximal label is less than 5 since s(D)>1. 

∙ When c(P)=9, d(P)=3 or 4 by Lemmas <ref>, <ref>. 
There are no diagrams D such that d(D)>4 by Lemma <ref>. 
When d(P)=3, each oriented alternating diagram D^A has warping degree labeling { 3, 4 } or { 5, 6 }. 
There are no nonalternating diagrams D such that the minimal label is greater than 3 and the maximal label is less than 6 since s(D)>1. 









 §.§ Welded unknotting number




In this subsection, we apply results in Subsection <ref> to a study of welded knot diagrams. 
A welded knot diagram is a diagram obtained from a classical knot diagram by replacing some classical crossings with “welded crossings”. 
A welded knot (<cit.>) is the equivalence class of welded knot diagrams related by classical and virtual Reidemeister moves and the over forbidden move. 
The welded unknotting number, u_w (D), of a welded knot diagram D is the minimum number of classical crossings which are needed to be replaced with a welded crossing to obtain a diagram of the trivial knot (<cit.>). 
Let D be a classical knot diagram. 
It is known that u_w (D) ≤d(D) (<cit.>), as a relation between the welded unknotting number and the warping degree. 
On the other hand, the inequality I(P) ≤ d(P) ≤ c(P)-I(P), the relation between the warping degree and the isolate-region number, was shown in <cit.>. 
For a knot diagram D, let I(D)=I(P), where P=|D|. 
With Propositions <ref>, <ref>, we have the following corollaries which are estimations of the welded unknotting number in terms of isolated regions. 



Let D be an almost alternating diagram. 
We have u_w (D) ≤ c(D)-I(D). 




Let D be an irreducible knot diagram of c(D) ≤ 9 such that |D| is neither of the projections in Figure <ref>. 
We have u_w (D) ≤ c(D)-I(D). 










§ REGION-CONNECT GRAPH AND ITS COMPLEMENT




In this section, region-connect graph and region-disconnect graph are defined and investigated. 
Then an algorithm to find the isolate-region number is given. 
For a graph G, we say a graph G' is a subgraph of G and denote by G' ⊆ G if G'=G or G' is obtained from G by deleting some vertices and edges.
For a link projection L with n regions, give labels x_1, x_2, … , x_n to the regions. 
The region-connect graph G of L is the graph with the vertices labeled x_1, x_2, … ,x_n and the edges x_α x_β for all the pairs of connected regions x_α and x_β (see Figure <ref>).
The region-disconnect graph, G, of L is the complement graph of G. 
By definition, G represents which pairs of regions are disconnected. 
Then we can find the isolated-region sets R_i of L by looking at G as follows. 
Find a subgraph of G which is a complete graph K_r. 
Since the corresponding r regions to the vertices of K_r are disconnected each other in L, the set of them is isolated. 
Moreover, the maximum size of a complete graph K_r ⊆G is identical to the isolate-region number I(K). 
For example, we obtain I(K)=3 for the knot projection K in Figure <ref> because the graph G includes K_3 and does not include K_r for r ≥ 4. 










§ I-GENERATING FUNCTION




In this section, we explore the generating functions for isolate-region sets of a link projection. 
Let L be a link projection. 
Let R= { x_1, x_2, … ,x_n } be the set of all regions of L. 
Let P(R) be the power set of R. 
Now we want to know how isolated-region sets R_i are distributed in P(R). 
First, each element R_k ∈ P(R) is determined by the choices whether each region x_j is taken or not. 
Hence, all the elements in P(R) can be seen as the terms of the following polynomial. 


    h    =(x_1^0 +x_1^1)(x_2^0 +x_2^1)(x_3^0 +x_3^1) ⋯
       =(1+x_1)(1+x_2)(1+x_3) ⋯
       = 1 + x_1 + x_2 + ⋯ +x_1 x_2 + x_1 x_3 + ⋯ + x_1 x_2 x_3 + ⋯ ,



where j=1 or 0 in x_i^j denotes if the region x_i is chosen or not respectively. 
For h, delete the terms which include the pair of connected regions, i.e., the pair of the letters which are connected in G. 
Then we obtain a polynomial g which represents all the isolated-region sets. 
Since we want to know just the number of isolated-region sets, replace each x_i with x. 
Then, we obtain the generating function f(L) for # (R_i) for isolated-region sets R_i. 
In this paper, we call f(L) the i-generating function of L. 
By definition, the maximum degree, maxdegf(L), indicates the isolate-region number of L. 
We have the following proposition. 




For each i-generating function f(L)=a_0 +a_1 x +a_2 x^2 +a_3 x^3 + … + a_r x^r of a connected link projection L with c crossings, a_0 =1 and a_1 = c+2. 
All the terms from degree 0 to maxdegf(L)=r are nonzero. 






We notice that each coefficient a_k is the number of isolated-region sets consisting of k regions. 
In terms of the region-disconnect graphs G, a_k is the number of K_k with K_k ⊆G. 
The first claim is obvious from the fact that we have one ∅ and the number of regions of L is c+2. 
For the second claim, when G includes a complete graph K_r, then G also includes K_r-1 because K_r ⊇ K_r-1. 




For connected irreducible link projections, we have an upper and lower bounds for a_2 as follows[We note that Proposition <ref> does not include Corollary <ref> because the proposition claims just a_2 ≥ (c^2 -5c+2)/2=-2 when c=4.].




Let L be a connected irreducible link projection with c crossings and let f(L)=a_0 + a_1 x + a_2 x^2 + … + a_r x^r be the i-generating function of L. We have 

    c^2 -5c+2/2≤ a_2 ≤c^2 +3c+2/2.






The upper bound is c+22, the number of edges of K_c+2 since G⊆ K_c+2. 
For the lower bound, let x_1, x_2, … , x_c+2 be the regions of L. 
Suppose x_i is a k_i-gon. 
Then x_i has at most 2 k_i regions around x_i. 
For each region x_i, the number of regions which is disconnected to x_i is at least (c+2)-2k_i -1, where c+2 implies the number of all regions of L. 
In total, we have 

    a_2 ≥∑_i=1^c+2c+1-2k_i/2 = (c+1)(c+2)/2- ∑_i=1^c+2k_i,

where we divided the summation by 2 to fix the double counting. 
Since we have 

    c= 2C_2 +3C_3 + 4C_4 + …/4 = 1/4∑_i=1^c+2 k_i

for connected irreducible link projections, we have 

    a_2 ≥c^2 +3c+2/2-4c = c^2 -5c +2/2.












§ I-GENERATING FUNCTIONS OF (2,N)-TORUS LINK PROJECTIONS




In this section, we investigate the i-generating functions for the standard projections of (2,n)-torus links, and prove the following theorem. 





The i-generating function f(L) of a standard projection  L of the (2,n)-torus link ( n ≥ 4 ) is

    f(L) =1+(n+2)x+ ∑ _k=2^⌊n/2⌋n-kkn/n-k x^k .







For a standard projection T(2,n) of a (2,n)-torus link, give labels of regions x_1, x_2, … , x_n+2 as shown in Figure <ref>.





We notice that the (n+1)th and (n+2)th regions are connected with all the other regions. 
Also, when n ≥ 3, each region from the 1st to nth is connected to the four regions, the (n+1)th and (n+2)th regions and the neighboring two bigons. 
In the region-connect graph G, therefore, the (n+1)th and (n+2)th vertices have degree (n+1), and the others have degree four. 
As for G, the (n+1)th and (n+2)th vertices have degree zero, and the others have degree (n-3). 
Moreover, in G, the vertices x_1, x_2, … , x_n forms a complete graph K_n missing the following n edges, x_1 x_2, x_2 x_3, x_3 x_4, … , x_n x_1. 
For example, when n=5, we have G and G as shown in Figure <ref>, and the polynomials h=(1+x_1)(1+x_2) … (1+x_7), g= 1+ x_1 +x_2 + … + x_7 + x_1 x_3 + x_3 x_5 +x_5 x_2 + x_2 x_4 + x_4 x_1, and f=1+7x+5x^2. 



To prove Theorem <ref>, we prepare a (perhaps well-known) formula. 




Suppose there are m white balls in a line. 
Denote by ( mn) the number of cases to paint n of m balls red so that the painted balls are not next to each other. Then, 

    ( mn) = m-n+1n.








At first, suppose (m-n) white balls in a line. 
Then there are (m-n+1) rooms between balls or on a side. 
The number of ways to put the painted n red balls there is m-n+1n.





Now we show Theorem <ref>. 



Proof of Theorem <ref>. 
Let G be the region-disconnected graph of T(2,n) with the (n+2) regions labeled as in Figure <ref>. 
Note that the (n+1)th and (n+2)th vertices have degree zero. 
For the subgraph of G consisting of the vertices x_1, x_2, … , x_n and the edges, we discuss how many complete graphs K_k are included. 

Let f=a_0 +a_1 x + a_2 x^2 + … + a_r x^r be the i-generating function. 
For the maximum degree, we have r= maxdegf= ⌊n/2⌋ because we can take ⌊n/2⌋ non-neighboring vertices and cannot for ⌊n/2⌋ +1 from G, where we say x_i and x_j are neighboring when |i-j|=1 or |i-j|= n-1. 

For coefficients, we have a_0 = 1 as the empty set of vertices. 
We have a_1 = n+2, the number of vertices (or K_1) of G, as mentioned in Proposition <ref>. 
We have a_2 = (n-3) n/2, the number of edges (or K_2) of G. 
More precisely, fix a vertex x_i. 
Then x_i has n-3 choices of vertices to form an edge in G. 
Remark that x_i cannot be connected with the neighboring vertices. 
Multiple n-3 by n because we can consider the same thing for the n vertices. 
Divide it by two since each edge is double counted. 
Hence, we have a_2 = (n-3) n/2. 

Next, for k ≥ 3, we obtain a_k = ( n-3k-1) n/k in the following way. 
Fix a vertex x_i. 
Then x_i has n-3 non-neighboring edges and has ( n-3k-1 ) choices to take other k-1 vertices to form K_k so that any pair of vertices is not neighboring. 
Multiple ( n-3k-1 ) by n, and divide it by k since each K_k is counted k times. 
Then, 


    a_k    = ( n-3k-1) n/k = (n-3)-(k-1)+1k-1n/k = n-k-1k-1n/k
       = n-k/n-k(n-k-1)(n-k-2) … (n-2k+1)/(k-1)!n/k = n-kkn/n-k.


This holds when k=2, too. 





The i-generating functions f_n of T(2,n) are listed below for n=1 to 12. 


    f_1    = 1 + 3x 
    
    f_2    = 1 + 4x 
    
    f_3    = 1 + 5x 
    
    f_4    = 1 + 6x + 2x^2 
    
    f_5    = 1 + 7x + 5x^2 
    
    f_6    = 1 + 8x + 9x^2 + 2x^3 
    
    f_7    = 1 + 9x + 14x^2 + 7x^3 
    
    f_8    = 1 + 10x + 20x^2 + 16x^3 + 2x^4 
    
    f_9    = 1 + 11x + 27x^2 + 30x^3 + 9x^4 
    
    f_10   = 1 + 12x + 35x^2 + 50x^3 + 25x^4 + 2x^5 
    
    f_11   = 1 + 13x + 44x^2 + 77x^3 + 55x^4 + 11x^5 
    
    f_12   = 1 + 14x + 54x^2 + 112x^3 + 105x^4 +36x^5 +2x^6





We have the following formula. 





For the i-generating functions f_n (x) of T(2,n), we have 

    f_n - f_n-1 - x f_n-2 = -2x^2

for n ≥ 4. 







Let b_k be the coefficient of x^k of f_n - f_n-1 - x f_n-2. 
When k ≥ 3, the kth coefficient of f_n is n-kkn/n-k, that of f_n-1 is n-1-kkn-1/n-1-k, and the (k-1)th coefficient of f_n-2 is (n-2)-(k-1)k-1n-2/(n-2)-(k-1) = n-k-1k-1n-2/n-k-1. 
Hence, 

    b_k = n-kkn/n-k - n-k-1kn-1/n-k-1 - n-k-1k-1n-2/n-k-1.


Using the formula n+1r= nr-1 + nr, we have n-kk = n-k-1k-1 + n-k-1k and 


    b_k    = n-k-1k-1( n/n-k - n-2/n-k-1) + n-k-1k( n/n-k - n-1/n-k-1) 
       = (n-k-1)!/(k-1)!(n-2k)!n-2k/(n-k)(n-k-1) - (n-k-1)!/k! (n-2k-1)!k/(n-k)(n-k-1)
       = (n-k-1)!/(k-1)!(n-2k-1)!1/(n-k)(n-k-1)- (n-k-1)!/(k-1)!(n-2k-1)!1/(n-k)(n-k-1)
       = 0.


For k ≤ 2, b_2 = n-22n/n-2 - n-32n-1/n-3 -n =-2, b_1 = (n+2)-(n+1)-1=0 and b_0 = 1-1=0. 
Hence, we have b_2 = -2 and b_k =0 for k ≠ 2, and f_n - f_n-1 - xf_n-2 = -2 x^2 holds for n ≥ 4. 






From Proposition <ref>, we have a recurrence relation about the number of isolated-region sets. 


For the i-generating function f_n (x) of T(2,n) of n ≥ 4, we have 

    f_n (1)=f_n-1(1)+ f_n-2(1)-2.










§ ACKNOWLEDGMENT


The authors thank Professor Rama Mishra for her support and encouragement.
The second author's work was partially supported by JSPS KAKENHI Grant Number JP21K03263. 



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