Antipodal Point Arrangements on Spheres and Classification of Normal Systems
C. P. Anil Kumar
math.CO Primary 52C35

Antipodal Point Arrangements on Spheres]
Antipodal Point Arrangements on Spheres and Classification of Normal Systems
C.P. Anil Kumar]Author: C.P. Anil Kumar*
Center for Study of Science, Technology and Policy
	# 18 & #19, 10th Cross, Mayura Street,
	Papanna Layout, Nagashettyhalli, RMV II Stage,
	Bengaluru - 560094
	Karnataka,INDIA
akcp1728@gmail.com
*The author is supported by a research grant and facilities provided by Center for study of Science, 
	Technology and Policy (CSTEP), Bengaluru, INDIA for this research work.
[2010]Primary: 52C35


For any positive integer k>1, we classify the antipodal point arrangements
on the sphere S^k up to an isomorphism,
by associating a finite complete set of cycle invariants.

[
    [
    December 30, 2023
=====================





§ INTRODUCTION

The main motivation to write this article, arises during the association of invariants such as normal systems to the infinity type hyperplane arrangements which is done in C. P. Anil Kumar <cit.>. The infinity type line arrangements, their nomenclature and some of their properties are discussed in C. P. Anil Kumar <cit.>.
Here we classify normal systems combinatorially. Before we restate the relevant Problem <ref> regarding classification of normal systems we need a few definitions.
 
	
	 

	Let N={L_1,L_2,…,L_n} be a finite set of lines passing through the origin in R^m. Let U={± v_1,± v_2,…,± v_n} be a set of antipodal pairs of non-zero vectors on these lines. We say that
	N forms a normal system, if the set 
	B={v_1,v_2,…,v_n}
	of vectors has the property that, any subset CB of cardinality at most m is a linearly independent set. 


	
	 

	Let  N_1={L_1,L_2,…,L_n},N_2={M_1,M_2,…,M_n} 
	be two finite sets of lines passing through the origin in R^m, both of them have the same 
	cardinality n, which form normal systems. Let 
	U_1={±v_1,±v_2,…,±v_n},U_2={±w_1,±w_2,…,±w_n} 
	be two sets of antipodal pairs of vectors on these lines in N_1,N_2 respectively. We say a bijection 
	:U_1U_2 is a convex positive bijection if 
	(-u)=-(u),u∈U_1 and for any basis B={u_1,u_2,…,u_m}U_1
	and a vector u∈U_1 we have 
	u   =i=1m∑a_iu_i  with a_i>0, 1≤i ≤m,  if and only if ,

		(u)   =i=1m∑b_i(u_i)  with b_i>0, 1≤i ≤m.
	We say two normal systems are isomorphic if there exists a convex positive bijection between their 
	corresponding sets of antipodal pairs of vectors. 

Now we mention the relevant open problem regarding classification of normal systems.
[Classification of Normal Systems and Finding Representatives in Each Isomorphism Class]

 

Let n,m be positive integers.
Classify and enumerate the normal systems in R^m up to an isomorphism by associating invariants which can be used to 
easily construct a family of normal systems representing each isomorphism class for every positive integer 
cardinality n of the normal system.

Here in this article we classify normal systems up to an isomorphism by associating a finite complete 
set of cycle invariants. The enumeration problem of the number of 
isomorphism classes of normal systems and the problem of representing their isomorphism classes 
by a well defined list of representatives still remain open (refer to Question <ref>) and known only for an initial few values of n for any m.
The exact statement of main Theorem <ref> about classification of normal systems cannot be stated here as it requires more definitions and concepts
which at present are not motivated and developed. Hence we defer the statement to its appropriate Section <ref> of the article.



 §.§ Brief Survey and the Structure of the Paper

With relevance to antipodal point arrangements (refer to Definition <ref>) or normal systems, the theory of matroids
is a well studied subject. Matroids are combinatorial abstractions of vector configurations and hyperplane arrangements.
Here in this article 
we study specific kind of antipodal pairs of vectors arranged on spheres, vector configurations,
which are associated to normal systems that arise
from hyperplane arrangements and classify them combinatorially. The method of associating cycle invariants
as a combinatorial model to point arrangements in 
the plane has already been explored by authors J.E.Goodman and R.Pollack <cit.>.
Also the slope problem mentioned in chapter 10, page 60 in M. Aigner and G. M. Ziegler <cit.>,
Proofs from THE BOOK, explains a similar method.

Section <ref> is devoted to the classification of antipodal point arrangements on S^2 in 
two dimensions. Theorem <ref> states the classification theorem in the case of dimension two.
Section <ref> revisits the two non-isomorphic examples of normal systems in dimension three
that are mentioned in C. P. Anil Kumar <cit.> and computes the combinatorial invariants.
Sections [<ref>-<ref>] are devoted to classification of antipodal point arrangements on S^k 
in higher dimensions for k>2. Theorem <ref> in Section <ref> states the classification theorem in higher dimensions. 
In final Section <ref> we pose open Question <ref> about normal systems.




§ ANTIPODAL POINT ARRANGEMENTS ON THE 2-SPHERE S^2


 

Now we define antipodal point arrangements on the 2-sphere S^2.


 

We say a set P_n={± P_1,± P_2,…,± P_n} S^2 of points is a point 
arrangement on the sphere if three points of P_n are linearly dependent then some two of them are antipodal.



 

Two point arrangements 
P_n={±P_1,±P_2,…,±P_n},Q_m={±Q_1,±Q_2,…,±Q_m} S^2 
are isomorphic if n=m and there is a bijection :P_n Q_n between the two sets such that the 
following occurs.


  * (-A)=-(A) for all A∈P_n.

  * for any A,B,C,D∈P_n if D is a positive combination of A,B,C if and only if 
(D) is a positive combination of (A),(B),(C). 

We sometimes also say that the isomorphism  is a convex positive bijection.
We say  is orientation preserving if for any three points A,B,C∈P_n the ordered triple (A,B,C) has positive determinant if and only if the ordered triple ((A),(B),(C)) has positive determinant. We say  is orientation reversing if for any three points A,B,C∈P_n the ordered triple (A,B,C) has positive determinant if and only if the ordered triple ((A),(B),(C)) has negative determinant.



Let P_n={±P_1,±P_2,…,±P_n},Q_n={±Q_1,±Q_2,…,±Q_n} S^2 
be two antipodal point arrangements on the 2-Sphere S^2. If :P_n Q_n is an isomorphism then it is either an orientation preserving isomorphism or it is an orientation reversing isomorphism. 


Let R_1,R_2,R_3∈P_n be linearly independent. Let S_1=(R_1),S_2=(R_2),S_3=(R_3)∈Q_n. We show that the sign of the product (R_1,R_2,R_3)(S_1,S_2,S_3) does not change for any such choice of three linearly independent elements R_1,R_2,R_3. If we change signs of R_i for some 1≤ i≤ 3 or interchange R_i,R_j for some 1≤ i≠ j≤ 3 and do the same operations with S_i,1≤ i≤ 3 then it is clear that the sign of product remains unchanged. It is also clear that if R∈P_n and R{± R_1,± R_2, ± R_3}
then there exists a,b,c∈R^* and d,e,f∈R^* such that we have 
   sign(a)=sign(d),sign(b)=sign(e),sign(c)=sign(f) and 
    R=aR_1+bR_2+cR_3,S=(R)=dS_1+eS_2+fS_3.
Hence again the sign of (aR_1+bR_2+cR_3,R_2,R_3)   (dS_1+eS_2+fS_3,S_2,S_3)=

   (R,R_2,R_3)(S,S_2,S_3) does not change. Here we have replaced R_1 by R and S_1 by S. 
This way we have that the sign does not change for any such choice of elements R_1,R_2,R_3 which proves the theorem.



 §.§ Algebraic Symbols Associated to a Four-Antipodal Point Arrangement on the Sphere S^2

We begin this section with the standard arrangement.


  §.§.§ The Standard Arrangement and its Associated Symbols


The arrangement S_4  S^2 consists of four antipodal pairs of points given by 
S_4={   x=[(1,0,0)],x=[(-1,0,0)],y=[(0,1,0)],-y=[(0,-1,0)],
    z=[(0,1,0)], -z=[(0,-1,0)],
P  (a point in  I-Octant),
   -P  (antipode point of  P  in  VII-Octant)} S^2
There are twenty four symbols that we associate to this standard arrangement. Before we actually describe these
symbols we mention four important aspects.


  * A symbol is of the form 
a(b,c,d)

  * We say that it is compatible or associated to an antipodal point arrangement if a,b,c,d represent elements
of the arrangement in S^2 such that a is a positive combination of b,c,d.

  * If we give an anticlockwise local orientation to the plane a^⊥ with the direction ray 
a∈ S^2 representing the thumb then ignoring signs the line cycle is clockwise oriented and is given by 
(bcd),  and not by (bdc).
For example to get the symbol P (y,x,z) for S_4 refer to the first octant view in Figure <ref>. Here the line cycle (yxz) is obtained by moving clockwise around P. 

  * 
The ordered triple (b,c,d) where a is a positive combination of b,c,d has negative determinant.

The associated symbols for the standard arrangement S_4 are given by  
   P (y,x,z), P (x,z,y),P (z,y,x),

   x(-y,P,-z),x(P,-z,-y),x(-z,-y,P),

   z(-x,P,-y),z(P,-y,-x),z(-y,-x,P),

   y(-z,P,-x),y(-x,-z,P),y(P,-x,-z),

   -x(-P,y,z),-x(z,-P,y),-x(y,z,-P),

   -z(-P,x,y),-z(x,y,-P),-z(y,-P,x),

   -y(-P,z,x),-y(x,-P,z),-y(z,x,-P),

   -P(-x,-y,-z),-P(-z,-x,-y),-P(-y,-z,-x).
In the above symbols the triples are all negatively (clockwise) oriented, that is, given that P is in the first octant
the triples have determinant negative. 



  §.§.§ The Symmetry Group on Four Elements and its Action on Symbols

 

Here we explore the symmetry involved in the above set of 24 compatible symbols. 
We state the following theorem on the action of the symmetry group S_4 on the set of symbols and 
describe the transitive orbits.


The group S_4 acts on the set 
S   ={p(q,r,s)|p,q,r,s ∈{±a,±b,±c,±d} such that 

   {±p}∩{±q}={±p}∩{±r}={±p}∩{±s}={±q}∩{±r}=

   {±q}∩{±s}={±r}∩{±s}=}
of all symbols with the action given by
p(q,r,s)  Apply  (12)  to get  -p (-r,-q,-s)

p(q,r,s)  Apply  (23)  to get  r (-q,p,-s)

p(q,r,s)  Apply  (34)  to get  s (-q,-r,p)

p(q,r,s)  Apply  (14)  to get  -p (-s,-r,-q).


  * The set S has 384 elements. Then each transitive orbit of an element under the action of S_4 contains 
24 elements. There are 16 orbits.

  * There are 8 orbits (192 elements satisfying property <ref>) 
that arise as compatible symbols associated to concrete four-antipodal point arrangements.

  * Each transitive orbit is the set of all compatible symbols corresponding to one fixed four antipodal pairs of 
points of the point arrangement on the sphere S^2 provided one of the symbols in the orbit is 
compatible.

  * Moreover the action of S_4 on the set S is free. 



We have #(S)=4*3!*2^4=384. We observe that the action is compatible with the relations
   (12)(23)(12)=(23)(12)(23),(23)(34)(23)=(34)(23)(34),

   (34)(14)(34)=(14)(34)(14),(12)(34)=(34)(12),(14)(23)=(23)(14),

   (12)^2=(23)^2=(34)^2=(41)^2=identity.
So we have an action of the symmetric group S_4 on the set S of symbols.
The set of compatible symbols, as a transitive orbit, obtained by the action of S_4 on the compatible
symbol P (y,x,z) is precisely the above given 24 compatible symbols of the standard arrangement
in Section <ref>. Similarly for every transitive orbit if one of the symbols is compatible then all 
the remaining 23 symbols of the orbit are compatible. The rest of the proof of the theorem is immediate.



  §.§.§ The Standard Arrangement and the Dictionary of Line-Cycles

Here we associate line cycles to the points of the standard arrangement S_4. 
Later we use this as a local dictionary for an antipodal arrangement on S^2 to characterize the arrangement up to an isomorphism.

Consider the standard four-antipodal point arrangement S_4  given by 
P_1   =x=[(1,0,0)],-P_1=-x=[(-1,0,0)],

P_2   =y=[(0,1,0)],-P_2=-y=[(0,-1,0)],

P_3   =z=[(0,1,0)],-P_3=-z=[(0,-1,0)],

P_4   =P  a point in  I-Octant,

-P_4   =-P  its antipode point in  VII-Octant  in S^2
The compatible 24 symbols (an S_4 transitive orbit) gives rise to the following dictionary of line cycles using subscripts {1,2,3,4} and symbols {+,-} at 
each point of the arrangement with (x,y,z) denoting a positively oriented basis of the arrangement.
   _4^+=(213) at P_4,_4^-=(231) at -P_4,

   _3^+=(142) at P_3,_3^-=(124) at -P_3,

   _2^+=(341) at P_2,_2^-=(314) at -P_2,

   _1^+=(243) at P_1,_1^-=(234) at -P_1.

Now we prove a theorem that given the dictionary of line cycles
there is a unique way to recover back the 24 compatible
symbols, an S_4 orbit of the arrangement, which is compatible with the standard arrangement.

We state the theorem as follows.


Let P_4={± P_1,± P_2,± P_3,± P_4} S^2 be any four-antipodal point 
arrangement on the sphere. Suppose the line cycles are given by
   _4^+=(213) at P_4,_4^-=(231) at -P_4,

   _3^+=(142) at P_3,_3^-=(124) at -P_3,

   _2^+=(341) at P_2,_2^-=(314) at -P_2,

   _1^+=(243) at P_1,_1^-=(234) at -P_1.
Then the map : P_4S_4 given by 
:   P_1x,-P_1-x,P_2y,-P_2-y,

   P_3z,-P_3-z,P_4P,-P_4-P
is an isomorphism, that is, it is a convex positive bijection. Also - is an isomorphism.
The S_4 invariant set of 24 compatible symbols are given by
P_4    (P_2,P_1,P_3), P_4 (P_1,P_3,P_2),P_4 (P_3,P_2,P_1),

P_1   (-P_2,P_4,-P_3),P_1(P_4,-P_3,-P_2),P_1(-P_3,-P_2,P_4),

P_3   (-P_1,P_4,-P_2),P_3(P_4,-P_2,-P_1),P_3(-P_2,-P_1,P_4),

P_2   (-P_3,P_4,-P_1),P_2(-P_1,-P_3,P_4),P_2(P_4,-P_1,-P_3),

-P_1   (-P_4,P_2,P_3),-P_1(P_3,-P_4,P_2),-P_1(P_2,P_3,-P_4),

-P_3   (-P_4,P_1,P_2),-P_3(P_1,P_2,-P_4),-P_3(P_2,-P_4,P_1),

-P_2   (-P_4,P_3,P_1),-P_2(P_1,-P_4,P_3),-P_2(P_3,P_1,-P_4),

-P_4   (-P_1,-P_2,-P_3),-P_4(-P_3,-P_1,-P_2),

-P_4   (-P_2,-P_3,-P_1).


Let us denote 
{P_1   =x,-P_1=-x,P_2=y,-P_2=-y,
P_3   =z,-P_3=-z,P_4=P,-P_4=-P}.
The octant views are given in Figure <ref> based on the point P lying in various octants
with respect to a positively oriented system (x,y,z).

However first we show that (P_1,P_2,P_3) is positively oriented, that is, its determinant is positive
and the symbol P_4 (P_2,P_1,P_3) is compatible. A priori we do not know the orientation of (P_1,P_2,P_3)
and the compatibility signs of the symbols.

Consider the following choices. (± P_1,± P_2,± P_3). Out of these 
(P_1,P_2,P_3),(-P_1,-P_2,P_3),(P_1,-P_2,-P_3),(-P_1,P_2,-P_3)
have the same sign of the determinant and the remaining
(-P_1,-P_2,-P_3),(P_1,P_2,-P_3),(-P_1,P_2,P_3),(P_1,-P_2,P_3)
have the same sign of the determinant.

If the second set of determinants are positive then we argue as follows using Theorem <ref>.
Suppose P_4 (-P_2,-P_1,-P_3) is compatible then we have -P_1 (P_2,P_4,P_3) is compatible.
Hence _1^-=(243) which is invalid.
Suppose P_4 (P_2,P_1,-P_3) is compatible then we have -P_3 (-P_1,P_4,-P_2) is compatible.
Hence _3^-=(142) which is invalid.
Suppose P_4 (P_2,-P_1,P_3) is compatible then we have -P_1 (-P_2,P_4,-P_3) is compatible.
Hence _1^-=(243) which is invalid.
Suppose P_4 (-P_2,P_1,P_3) is compatible then we have -P_2 (-P_3,P_4,-P_1) is compatible.
Hence _2^-=(341) which is invalid.

If the first set of determinants are positive then we argue as follows using Theorem <ref>.
We have P_4 (-P_2,-P_1,P_3),P_4 (-P_2,P_1,-P_3),P_4 (P_2,-P_1,-P_3) also give invalid line cycles. 
Hence we conclude that (P_1,P_2,P_3) is positively oriented and the symbol P_4 (P_2,P_1,P_3)
is compatible.

This proves the theorem. We also note that the over all total flip given by 
: P_4S_4,:   P_1-x,-P_1x,P_2-y,-P_2y,

   P_3-z,P_3z,P_4-P,-P_4P
is also an isomorphism, that is, a convex positive bijection. Using these line cycles we can write down all the S_4 invariant set of
24 compatible symbols.


There are other isomorphisms from P_4 to S_4 as well and below we describe all of them via the automorphism 
group (S_4).



  §.§.§ Automorphism Group of the Standard Antipodal Point Arrangement

Here we compute the automorphism group of the standard antipodal point arrangement.

Let S_4 S^2 be the standard arrangement. Then 
(S_4)=S_4 ⊕(/2).


We have the twenty four compatible symbols of the standard arrangement given in Section <ref>.
If p (q,r,s) is one such compatible symbol then the map 
:S_4S_4,:   Pp,-P-p,yq,-y-q,

   xr,-x-r,zs,-z-s
is an automorphism. We also have if  is an automorphism then - is also an automorphism and
moreover either (P) ((y),(x),(z)) or -(P) (-(y),-(x),-(z)) gives rise to a compatible
symbol and the other one is not a compatible symbol. Hence we get 
(S_4)=S_4 ⊕(/2).
This proves the theorem.




 §.§ An Isomorphism Theorem for Antipodal Point Arrangements on the Two-Dimensional Sphere 
S^2

 

Now we prove an isomorphism theorem for antipodal point arrangements on the sphere S^2
which can be generalized to higher dimensions in Theorem <ref>.



  §.§.§ Localization to Antipodal Point Sub-arrangements

 

Let P_n={± P_1,± P_2,…, ± P_n} S^2 be an antipodal point arrangement.
We introduce an equivalence relation ∼ on the symmetric group S_n as follows. 
Let g,h∈ S_n then g∼ h if g=h or g=h^-1. This is an equivalence relation with reflexive, symmetric and
transitive properties. The equivalence classes being [,^-1]. Any element of order at most 
two is an equivalence class containing just one element. Remaining equivalence classes has two elements.
The antipode map a:S^2 S^2 (reflection about the origin in three dimensions) has a negative determinant. 
The line cycles _i^+ for P_i, _i^- for -P_i associated to a pair of antipodes in P_n
are mutually inverses of each other as there is a reflection about the origin
is involved. So we actually obtain 
(n-1)-cycles with _i^-= (_i^+)^-1∈S_n-1({1,2,…,i-1,i+1,…,n}),1≤i≤n.
Now we consider the local scenario by restricting to just four antipodal pairs.
The restriction map |_local_A and inverse map (*)^-1 commutes.
We observe that 
For any four subset A   {1,2,…,n},
 (_i^-)_|_local_A   =((_i^+)^-1)_|_local_A=
((_i^+)_|_local_A)^-1,i∈A


  §.§.§ The Main Isomorphism Theorem in Two Dimensions

Now we state the theorem as follows.


The following two assertions hold true.


  * The line cycles of the antipodal pairs of points of a point arrangement 
P_n S^2 determines the collection of local S_4-invariant
set of compatible symbols for every four subset of antipodal pairs of points in P_n.

  * Let 
P^1_n={± P^1_1, ± P^1_2,…,± P^1_n},P^2_n={± P^2_1,± P^2_2,…,± P^2_n}
be two point arrangements. Let (_i^+)_j be the line cycle associated to P^j_i and 
(_i^-)_j be the line cycle associated to -P^j_i for j=1,2,1≤ i≤ n.
There exists a convex positive bijection (an isomorphism) :P^1_nP_n^2 if and only if there exist 


  * a permutation ∈ S_n and

  * a sign vector =((1),(2),…,(n))∈ (/2)^n={± 1}^n

with the property that 


  * 
either 
   (_(i)^[(i)*(+)])_2=(_i^+)_1^-1,

   (_(i)^[(i)*(-)])_2=(_i^-)_1^-1,
1≤i≤n

  * 
or an overall total flip (here we can choose - in place of )
   (_(i)^[(i)*(+)])_2=[(_i^+)_1^-1]^-1=(_i^-)_1^-1,

   (_(i)^[(i)*(-)])_2=[(_i^-)_1^-1]^-1=(_i^+)_1^-1,
1≤i≤n

where 


  * [(i)*(+ )]=+,[(i)*(-)]=- if (i)=+. 

  * [(i)*(+)]=-,[(i)*(-)]=+ if (i)=-.




We prove the second assertion first.
Suppose :P^1_nP_n^2 is an orientation preserving or orientation reversing isomorphism as it is an isomorphism using Theorem <ref>. Then the permutation  and the signed vector 
are defined by the equation
(P_i)=(i)P_(i),(-P_i)=-(i)P_(i),1≤i≤n.
Now with this definition of , the property <ref> is satisfied if  is orientation preserving and the property <ref> is satisfied if it is orientation reversing.
If we choose for  the following definition
(P_i)=-(i)P_(i),(-P_i)=(i)P_(i),1≤i≤n
then , satisfies the property <ref> if  is orientation preserving and the property <ref> is satisfied if it is orientation reversing. This proves one way implication. 

Now we prove the other way implication where we are given the permutation  and the signed vector  and changing 
to - if necessary we assume that the property <ref> holds. 
First we localize to any two corresponding four-antipodal point arrangements
{±P_i,±P_j,±P_k,±P_l},{±P_(i),±P_(j),±P_(k),±P_(l)}.
Since property <ref> holds and the restriction map and the inverse map commutes with respect to localization 
there is an isomorphic way to identify these two arrangements
using local line cycles via the local chart as the standard arrangement
S_4 using Theorem <ref>. Using this chart we conclude that locally there exists an isomorphism
of the four-antipodal point arrangements given by 
:   P_i(i)P_(i),-P_i-(i)P_(i),P_j(j)P_(j),-P_j-(j)P_(j)

   P_k(k)P_(k),-P_k-(k)P_(k),P_l(l)P_(l),-P_l-(l)P_(l)
These local isomorphisms patch up and extend uniquely to an isomorphism defined as 
(P_i)=(i)P_(i),(-P_i)=-(i)P_(i),1≤i≤n.
We also observe that -:P^1_nP^2_n is an isomorphism. Moreover it is either orientation preserving or orientation reversing.
This proves the isomorphism theorem in two dimensions.

Now we prove the first assertion. The local cycles of four-antipodal subarrangements determine the S_4 invariant set of 24
compatible symbols using Theorem <ref>. 
Hence the first assertion follows and we can write down all the compatible symbols of the given arrangement.



§ EXAMPLES OF TWO NON-ISOMORPHIC NORMAL SYSTEMS IN THREE DIMENSIONS OVER RATIONALS: REVISITED


Consider the normal systems whose associated sets of antipodal vectors are given by   
U_1={±u_i|1≤i ≤6},U_2={±v_i|1≤i ≤6}
with U_1∩U_2={± u_1,± u_2,± u_3,± u_4,± u_5}={± v_1,± v_2,± v_3,± v_4,± 
v_5} where
   u_1=(1,0,0)=v_1,u_2=(0,1,0)=v_2,u_3=(0,0,1)=v_3,

   u_4=(1/3,2/3,2/3)=v_4, u_5=(1/9,4/9,8/9)=v_5,
u_6=(6/11,6/11,7/11),v_6=(2/11,6/11,9/11).
Here below we find out line cycles of each point with respect to the given notation.

We have proved that these two are non-isomorphic normal systems by associating graphs of compatible pairs 
mentioned in C. P. Anil Kumar <cit.>.
For example, from the 15 equations below for U_1 the vertex {-u_1,u_2} has degree one and 
is only compatible with {u_4,-u_6} (equation (5) in the first set). From the 15 equations below for U_2 we observe that there
is no vertex of degree one as we observe that if a vertex of the associated graph of
compatible pairs has a positive degree then the degree is at least two. 

Now we mention the following 64=15 equations for U_1.


  * 3u_4= u_1+2u_2+2u_3=(1,2,2).

  * 9u_5=u_1+4u_2+8u_3=(1,4,8).

  * 11u_6=6u_1+6u_2+7u_3=(6,6,7).

  * 12u_4=3u_1+4u_2+9u_5=(4,8,8).

  * 5u_1+21u_4=2u_2+22u_6=(12,14,14).

  * 88u_6=41u_1+20u_2+63u_5=(48,48,56).

  * u_1+9u_5=4u_3+6u_4=(2,4,8).

  * 11u_6=3u_1+u_3+9u_4=(6,6,7).

  * 9u_1+9u_5=10u_3+22u_6=(12,12,24).

  * 9u_5=2u_2+6u_3+3u_4=(1,4,8).

  * 18u_4=6u_2+5u_3+11u_6=(6,12,12).

  * 54u_5=18u_2+41u_3+11u_6=(6,24,48).

  * 44u_6=13u_1+30u_4+9u_5=(24,24,28)

  * 123u_4=26u_2+45u_5+66u_6=(41,82,82).

  * 13u_3+27u_4=27u_5+11u_6=(9,18,31).

Then we have by actual computation the line cycles are given as 
   (_1^+)_1=(24653)   at u_1,(_1^-)_1=(23564)  at -u_1,

   (_2^+)_1=(13546)  at u_2,(_2^-)_1=(16453)  at -u_2,

   (_3^+)_1=(16452)  at u_3,(_3^-)_1=(12546)  at -u_3,

   (_4^+)_1=(15326)  at u_4,(_4^-)_1=(16235)  at -u_4,

   (_5^+)_1=(16432)  at u_5,(_5^-)_1=(12346)  at -u_5,

   (_6^+)_1=(15324)  at u_6,(_6^-)_1=(14235)  at -u_6.

Now we mention the following 64=15 equations for U_2.


  * 3v_4=v_1+2v_2+2v_3=(1,2,2).

  * 9v_5=v_1+4v_2+8v_3=(1,4,8).

  * 11v_6=2v_1+6v_2+9v_3=(2,6,9).

  * 12v_4=3v_1+4v_2+9v_5=(4,8,8).

  * 27v_4=5v_1+6v_2+22v_6=(9,18,18).

  * 88v_6=7v_1+12v_2+81v_5=(16,48,72).

  * v_1+9v_5=4v_3+6v_4=(2,4,8).

  * v_1+11v_6=3v_3+9v_4=(3,6,9).

  * v_1+27v_5=6v_3+22v_6=(4,12,24).

  * 9v_5=2v_2+6v_3+3v_4=(1,4,8).

  * 11v_6=2v_2+5v_3+6v_4=(2,6,9).

  * 18v_5=2v_2+7v_3+11v_6=(2,8,16).

  * v_1+44v_6=18v_4+27v_5=(9,24,36).

  * 66v_6=2v_2+21v_4+45v_5=(12,6,54).

  * v_3+11v_6=3v_4+9v_5=(2,6,10).


Then we have by actual computation the line cycles are given as 
   (_1^+)_2= (24653)  at v_1,(_1^-)_2= (23564)  at -v_1,

   (_2^+)_2= (13564)  at v_2,(_2^-)_2= (14653)  at -v_2,

   (_3^+)_2= (14652)  at v_3,(_3^-)_2= (12564)  at -v_3,

   (_4^+)_2= (16532)  at v_4,(_4^-)_2= (12356)  at -v_4,

   (_5^+)_2= (14632)  at v_5,(_5^-)_2= (12364)  at -v_5,

   (_6^+)_2= (14532)  at v_6,(_6^-)_2= (12354)  at -v_6.

As an example we just show that (_1^+)_1=(24653). For this we observe the following. 


  * u_1=-2u_2-2u_3+3u_4 and 0-1012200-1<0. Hence the compatible cycle is (243).

  * u_1=-4u_2-8u_3+9u_5 and 0-1014800-1<0. Hence the compatible cycle is (253).

  * 6u_1=-6u_2-7u_3+11u_6 and 0-1066700-1<0. Hence the compatible cycle is (263).

  * 3u_1=-4u_2+12u_4-9u_5 and 0-10122-1-4-8<0. Hence the compatible cycle is (245).

  * 5u_1=2u_2-21u_4+22u_6 and 010-1-2-2667<0. Hence the compatible cycle is (246).

  * 41u_1=-20u_2-63u_5+88u_6 and 010667-1-4-8<0. Hence the compatible cycle is (265).

  * u_1=4u_3+6u_4-9u_5 and 001122-1-4-8<0. Hence the compatible cycle is (345).

  * 3u_1=-u_3-9u_4+11u_6 and 00-1-1-2-2667<0. Hence the compatible cycle is (346).

  * 9u_1=10u_3-9u_5+22u_6 and 001667-1-4-8<0. Hence the compatible cycle is (365).

  * 13u_1=-30u_4-9u_5+44u_6 and -1-2-2667-1-4-8<0. Hence the compatible cycle is (465).

Hence the 5-cycle compatible with all the ten 3-cycles is given by (24653).
The computation of the rest of the five cycles is similar. 


§ ANTIPODAL POINT ARRANGEMENTS ON HIGHER DIMENSIONAL SPHERES AND CLASSIFICATION OF NORMAL SYSTEMS


Here we mainly associate combinatorial invariants to antipodal point arrangements to classify them and hence classify
the normal systems up to an isomorphism. These combinatorial invariants turn out to be oriented cycles of
points of the orthogonally projected arrangements along small sub-arrangements. We begin with the required definitions.


 

We say a set P_n={± P_1,± P_2,…,± P_n} S^k of points is a point 
arrangement on the sphere if for any 1≤ i_1<i_2<…<i_k+1≤ n the points 
P_i_1,P_i_2,…,P_i_k+1 are linearly independent.



 

Two point arrangements 
P_n={±P_1,±P_2,…,±P_n},Q_m={±Q_1,±Q_2,…,±Q_m} S^k 
are isomorphic if n=m and there is a bijection :P_n Q_n between the two sets such that the 
following occurs.


  * (-A)=-(A) for all A∈P_n.

  * for any A, A_i_l∈P_n,1≤ l≤ k+1,A is a positive combination of A_i_l,1≤ l≤ k+1
if and only if (A) is a positive combination of (A_i_l),1≤ l≤ k+1.

We sometimes also say that the isomorphism  is a convex positive bijection.
We say  is orientation preserving if for any k+1 points A_i_l∈P_n,1≤ l≤ k+1 the ordered tuple (A_i_1,…,A_i_l) has positive determinant if and only if the ordered tuple ((A_i_1),…,(A_i_l)) has positive determinant. We say  is orientation reversing if for any k+1 points A_i_l∈P_n,1≤ l≤ k+1 the ordered tuple (A_i_1,…,A_i_l) has positive determinant if and only if the ordered tuple ((A_i_1),…,(A_i_l)) has negative determinant.




Let P_n={±P_1,±P_2,…,±P_n},Q_n={±Q_1,±Q_2,…,±Q_n} S^k
be two antipodal point arrangements on the sphere S^k. Let :P_n Q_n be an isomorphism. Then it either an orientation preserving isomorphism or an orientation reversing isomorphism.


The proof is similar to that of Theorem <ref>.



§ DIMENSION REDUCTION AND MULTIPLE ORTHOGONALLY PROJECTED ANTIPODAL ARRANGEMENTS ALONG SMALL SUB-ARRANGEMENTS

 

We begin with a definition.

 

Let P_n={± P_1,… ,± P_n} be an antipodal point arrangement in the k-dimensional sphere 
S^k. Let A={± P_i_1,± P_i_2,…,± P_i_r} P_n
be an antipodal point sub-arrangement with 1≤ r≤ k-2. 
We can orthogonally project using Q^A the sub-arrangement P_nA to the space orthogonal 
to the space spanned by vectors of A to obtain an antipodal point arrangement 
P_n-r^A={P^A_j=Q^A(P_j),-P^A_j=Q^A(-P_j)
|1≤j≤n,j≠i_l,1≤l≤r} 
in the (k-r)-dimensional sphere S^k-r.

Now we prove a theorem on the signs.


 

Let P_n={± P_1,… ,± P_n} be an antipodal point arrangement in the k-dimensional sphere 
S^k. Let A={± P_n}. Let P^A_n-1={± P_1^A,± P_2^A,…,± P_n-1^A} denote the projected arrangement.
Suppose P_i=[(x^i_1,x^i_2,…,x^i_k,x^i_k+1)],-P_i=-[(x^i_1,x^i_2,…,x^i_k,x^i_k+1)],1≤i≤n. 
Suppose we have 
(x^j_1,x^j_2,…,x^j_k,x^j_k+1)=l=1k∑_l(x^i_l_1,x^i_l_2,…,x^i_l_k,x^i_l_k+1)
+_n(x^n_1,x^n_2,…,x^n_k,x^n_k+1)
for some j{i_1,i_2,…,i_k,n}, _l,_n∈R^*.
Suppose we have 
P_j^A=Q^A(x^j_1,x^j_2,…,x^j_k,x^j_k+1)=l=1k∑_lP^A_i_l=l=1k∑_lQ^A(x^i_l_1,x^i_l_2,…,x^i_l_k,x^i_l_k+1)
then sign(_l)=sign(_l),1≤l≤k.


This theorem is immediate.


Now we prove an isomorphism theorem about signs for antipodal point arrangements on spheres S^k.


 

Let P^j_n={± P^j_1,± P^j_2,…,± P^j_n} be two antipodal point arrangements in S^k for j=1,2. 
Let ±P^j_i_1,±P^j_i_2,…,±P^j_i_k+1,±P^j_l
be k+2 antipodal pairs of points of choice of the arrangement. Let A={i_1,i_2,…,i_k+1}. 
With respect to the set A let P^j_l=r=1m∑(^l_i_r)^j_AP^j_i_r,j=1,2.
Suppose we have sign((^l_i_r)^1_A)=sign((^l_i_r)^2_A) for any such choice. Then the map 
:P^1_iP^2_i,:-P^1_i-P^2_i, 1≤i≤n
is an isomorphism of antipodal point arrangements P^j_n.


This theorem is immediate and  is a convex positive bijection.



 §.§ Line Cycle Invariants Associated to Points of the Projected Arrangements

 

Let P_n={± P_1,± P_2,…,± P_n} be an antipodal
point arrangement in S^k. Let 
A={±P_j_1,±P_j_2,…,±P_j_k-2}P_n be a subset of cardinality k-2. Then 
consider the projected arrangement P^A_n-k+2 S^2. For various choices of A, these arrangements on the
two-dimensional spheres give rise to clockwise oriented line cycles at each point of P^A_n-k+2
denoted as follows:
(_j^+)^A   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at P^A_j,

(_j^-)^A   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at -P^A_j
both of which are (n-k+1)-cycles which are mutual inverses of each other for 1≤ j≤ n,j≠ j_l,1≤ l≤ k-2.



§ THE MAIN THEOREM


Now we prove the isomorphism theorem for the line cycle invariants.
[Main Theorem]
theorem:LCI
 

Let P_n={± P_1,± P_2,…,± P_n} be an antipodal point 
arrangement on S^k. The line cycle invariants of antipodal pairs 
P_j,-P_j,1≤j≤n given by mutually inverse cycles
(_j^+)^A   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at P^A_j,

(_j^-)^A   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at -P^A_j
after projection along the small sub-arrangement 
A={±P_j_1,±P_j_2,…,±P_j_k-2}P_n{±P_j} 
given by 
P_n-k+2^A={±P_i^A|i∈{1,2,…,n}{j_1,j_2,…,j_k-2}} 
for all such possible choices of A determine the 
antipodal point arrangement up to an isomorphism, that is,   
for l=1,2 let  
P^l_n={±P^l_1,±P^l_2,…,±P^l_n}S^k 
be two antipodal point arrangements with the line cycle invariants of antipodal pairs 
P^l_j,-P^l_j,1≤j≤n given by mutually inverse cycles
(_j^+)^A_l_l   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at (P^l_j)^A_l,

(_j^-)^A_l_l   ∈S_n-k+1({1,2,…,n}{j_1,j_2,…,j_k-2,j}) 
 at -(P^l_j)^A_l
after projection along the small sub-arrangement 
A_l={±P^l_j_1,±P^l_j_2,…,±P^l_j_k-2}P^l_n{±P^l_j},
then they are isomorphic by a convex positive bijection if and only if there exist  


  * a permutation ∈ S_n and 

  * a sign vector ∈ (/2)^n={±}^n

such that for all invariant line cycles either 
(^[(j)*(+)]_(j))^(A_1)_2 = (^+_j)^A_1_1^-1,

 which is equivalent to (^[(j)*(-)]_(j))^(A_1)_2 = (^-_j)^A_1_1^-1
holds or with a total flip the following holds. (Also we could flip the sign of )
(^[(j)*(-)]_(j))^(A_1)_2 = (^+_j)^A_1_1^-1,

 which is equivalent to (^[(j)*(+)]_(j))^(A_1)_2 = (^-_j)^A_1_1^-1
where 


  * (A_1)={± P^2_(j_1),± P^2_(j_2),…,± P^2_(j_k-2)}.

  * 

  * [(j)*(+)]=+,[(j)*(-)]=- if (j)=+.

  * [(j)*(+)]=-,[(j)*(-)]=+ if (j)=-.




	A convex positive bijection is either orientation preserving or orientation reversing using Theorem <ref>.
	The orientation preserving or the orientation reversing convex positive bijection  and the permutation ∈ S_n with a sign vector =((1),…,(n))∈ (/2)^n={±}^n are related by the following equations.
	For 1≤i≤n,(P^1_i)=(i)P^2_(i).
	We can assume if necessary and without loss of generality that  is trivial and =(+,…,+) so that (P^1_i)=P^2_i,1≤ i≤ n.
	If  is orientation preserving then we have 
	(^+_j)^A_2_2 = (^+_j)^A_1_1  and (^-_j)^A_2_2 = (^-_j)^A_1_1.
	If  is orientation reversing then we have 	
	(^-_j)^A_2_2 = (^+_j)^A_1_1  and (^+_j)^A_2_2 = (^-_j)^A_1_1.
	Here A_l={±P^l_j_1,±P^l_j_2,…,±P^l_j_k-2}P^l_n{±P^l_j},l=1,2.
	
To determine the arrangement up to an isomorphism we do the following. For r=1,2 let P^r_i_1,P^r_i_2,…,P^r_i_k+1,P^r_l
be k+2 points of the arrangements. With respect to A={i_1,…,i_k+1} define the coefficients (^l_i_j)^1_A,(^l_i_j)^2_A by  
letting P^1_l=j=1k+1∑(^l_i_j)^1_AP^1_i_j,
	P^2_l=j=1k+1∑(^l_i_j)^2_AP^2_i_j.
To determine the arrangement up to an isomorphism using Theorem <ref> 
we need to determine the signs of (^l_i_j)^1_A,(^l_i_j)^2_A,1≤ j ≤ k+1 
using the combinatorial invariants.

Now when k=2 we know that the line cycles give rise to the compatible set of S_4-invariant set of 24
symbols locally for all the sub-arrangements using Theorem <ref>. 
These determine the signs and hence the antipodal point arrangement on S^2 is determined 
up to an isomorphism.

Now we consider a general value of k. Now using various orthogonal projections along subsets of {1,…,n} 
and repeated application of Theorem <ref> we can recover the signs of the coefficients (^l_i_j)^1_A,(^l_i_j)^2_A 
from the combinatorial line cycle invariants. Now we use Theorem <ref>.

The rest of Theorem <ref> also follows.




§ OPEN QUESTIONS: THE ENUMERATION PROBLEM AND THE PROBLEM OF A COMPLETE LIST OF REPRESENTATIVES


We have solved the classification problem of isomorphism classes of normal systems in any dimension. Now we mention the two questions which are still open.



 

Let n,m be positive integers.


  * (Enumeration Problem): Enumerate the isomorphism classes of normal systems in R^m
of cardinality n.

  * (Representation Problem): Construct a complete list of representatives for the list of
isomorphism classes of normal systems in R^m of cardinality n.



abbrv
'
1
MR3823190 M. Aigner, G. M. Ziegler,
Proofs from THE BOOK, Sixth Edition, ISBN-13: 978-3-662-57264-1; 978-3-662-57265-8, Springer, Berlin, 2018, viii+326 pp, <https://doi.org/10.1007/978-3-662-57265-8>, MR3823190

CPAK C. P. Anil Kumar,
On the triangles in certain types of line arrangements, Discrete Mathematics, Algorithms and Applications, Online Published Nov. 2020, <https://doi.org/10.1142/S1793830921500312>, <https://arxiv.org/pdf/1906.05120.pdf>

NRHA C. P. Anil Kumar,
On Infinity Type Hyperplane Arrangements and Convex Positive Bijections: Preprint (2020), 24 pages, <https://arxiv.org/pdf/1711.07030.pdf> 

MR0583961 J. E. Goodman and R. Pollack,
On the Combinatorial Classification of Non-degenerate Configurations in the Plane, Journal of Combinatorial Theory Series A.  29(2), (1980), 220–235, ISSN 0097-3165, <https://doi.org/10.1016/0097-3165(80)90011-4>, MR0583961


