Graph Theoritical Approach: The Complete Set of ODCH Coxeter Polyhedra P10-1

Graph Theoritical Approach: The Complete Set of ODCH Coxeter Polyhedra P10-1

Graph Theoritical Approach:

The Complete Set of ODCH Coxeter Polyhedra P10-1

Dhrubajit Choudhury

The Director, HDR Foundation, H-19, Shankar Path, Hatigaon, Guwahati, Assam, India-781038,

1. A n-dimensional orbifold is a topological space with a structure based on the quotient space of n by a finite group action. An orbifold is called good if its universal cover is a manifold. We will concentrate only on good orbifolds.

   To give a hyperbolic structure on an orbifold, we model it locally by the orbit spaces of finite

   subgroups of PO 1, n acting on open subsets of Hn . Similarly, to put a real projective

   structure on an orbifold, we model it locally by the orbit spaces of finite subgroups of

   PGL n 1, acting on open subsets of Pn .

   A real projective structure on an orbifold M implies that M has a universal cover M and the

   deck transformation group 1

   M

   acting on M so that

   M is homeomorphic to M.

   1 M

   A convex set in Pn is a convex set in an affine patch. If we use Klein's model of a n-

   dimensional hyperbolic space, then is an open ball in Pn and PO 1, n is a subgroup of

   PGL n 1, preserving Hn . Therefore Hn can be imbedded in an n 1-dimensional real

   vector space V as an upper part of hyperboloid

   x2 x2 … x2

   1

   1 2 n1

   Hence hyperbolic orbifolds naturally have real projective structures. But a real projective structure of an orbifold may not have hyperbolic structure.

   We will concentrate on 3-dimensional compact hyperbolic orbifolds whose base spaces are homeomorphic to a convex polyhedron and whose sides are silvered and each edge is given

   an order. If the dihedral angle of an edge of a compact hyperbolic polyhedron is say that the order of the edge is n where n is a positive number.

   then we

   n

   Definition 1.0.1. Let X be S3 , E3

   , or

   H3 . Let Isom(X) denotes the group of isometries of

   X. A Coxeter polyhedron is a convex polytope in X whose dihedral angles are all integer sub- multiples of . Let P be a 3-dimensional Coxeter polyhedron and be the group generated by the reflections in the faces of P. Then is a discrete group of Isom(X) and P is its fundamental polyhedron. Conversely, every discrete group of Isom(X) can be obtained from a Coxeter polyhedron P such that P is its fundamental polyhedron. The number of faces intersect at vertex is called the degree of that vertex. Also the edge order of edges of a Coxeter polyhedron are positive integers.

   Definition 1.0.2. Let P be a fixed convex polyhedron. Let us assign orders at each edge. Let e

   be the number of edges and e2

   be the numbers of order-two. Let f be number of sides.

   We remove any vertex of P which has more than three edges ending or with orders of the edges ending there is not of the form

   2, 2, n, n 2,2,3,3,2,3, 4,2,3,5,

   1. ., orders of spherical triangular groups. This make P into an open 3-dimensional orbifold.

      Let

      P denote the differential orbifold with sides silvered and the edge orders realized as

      assigned from P with vertices removed. We say that

      P has a Coxeter orbifold structure.

      P of projective structures on an orbifold P

      is the

      space of all projective structures on

      P quotient by isotopy group actions of P .

      Definition 1.0.4. We say P is orderable if we can order the sides of P so that each sides meets sides meets sides of higher index in less than or equal to 3 edges.

      Example 1.0.5. Cube and dodecahedron are not satisfying orderability condition.

      be the orbifold structure of a 3-dimensional polyhedron P. We say

      that the orbifold structure

      P is orderable if the sides of P can be ordered so that each side

      has no more than three edges which are either of order 2 or included in a side of higher index.

      P is trivalent if each side F has three or less number of edges of order two or edges

      belonging to sides of higher class than F.

      Definition 1.0.7. A combinatorial polyhedron is a 3-ball whose boundary sphere S2 is

      equipped with a cell complex whose 0-cells, 1-cells and 2-cells will also be called vertices, edges and faces respectively, and which can be realized as a convex polyhedron. Topologically, a compact polyhedron P is a combinatorial polyhedron. A polyhedron is called trivalent if degree of each vertex is 3.

      Remark 1.0.8. For our convenient, we will use notation in short as:

      1. A compact hyperbolic Coxeter polyhedron is as CH-Coxeter polyhedron.

      2. An orderable and projectively deformable compact hyperbolic Coxeter orbifold is as ODCH-Coxeter orbifold.

   In my previous article [APPLICATION OF PLANTRI GRAPH: All COMBINATORIAL STRUCTURE OF ORDERABLE AND DEFORMABLE COMPACT COXETER

   HYPERBOLIC POLYHEDRA], we proved the following proposition.

   Proposition 1.0.9. Let P be a 3-dimensional CH-Coxeter polyhedron and P

   be its Coxeter

   orbifold structure. Suppose that P

   is orderable and projectively deformable. Then the total

   number of combinatorial polyhedral of such P is 5 and P is one of the combinatorial polyhedrons T, P6, P8-1, P10-1 and P10-2. The figure of P10-1 is as follows:

   In this article, we find the complete set of orderable and deformable compact Coxeter hyperbolic polyhedron P10-1.

   P10-1

   be its Coxeter orbifold structure. Suppose that

   P10-1

   is orderable and projectively

   deformable. Then the total number of such P10-1 is 5 and P10-1 is one of the polyhedrons in figure 2.

   Proof. Using Chois theorem, we prove that the number of edges of order 2 is exactly 5. By Andreevs condition for compact hyperbolic polyhedrons, we prove that the order of the

   edges at each vertex is one of the form 2,3,3,2,3, 4, 2,3,5 . Using graphical properties, we can assign the orders to each edge in five different ways up to symmetry. Then we assign the order of the faces in each of these five polyhedrons to form ODCH Coxeter polyhedron.

2. 1. Andreevs Theorem. In 1970, E.M. Andreev provides a complete characterization of 3- dimensional compact hyperbolic polyhedral having non-obtuse dihedral angles on his article [2]. Therefore Andreevs theorem is a fundamental tool for classification of 3 dimensional compact hyperbolic Coxeter polyhedron. Some elementary faces about polyhedral are essential before we state Andreevs theorem.

      Definition 2.1.1. A cell complex on S2 is called trivalent if each vertex is the intersection of

      three faces. A 3-dimensional combinatorial polyhedron is a cell complex C on S2

      satisfied the following condition:

      1. Every edges of C is the intersection of exactly two faces.

      2. Anon-empty intersection of two faces is either an edge or a vertex.

         that

      3. Every faces contains not fewer than 3 edges. If a face contains n edges then n is called the length of the face.

         Suppose C* be the dual complex of C in S2 . Then C* is a simplicial complex which embed

         in the same S2 so that the vertex correspond to face of C, etc. A simple closed curve in

         C* is called k-circuit if it is formed by k edges of C*. A k-circuit is called prismatic k- circuit if the intersection of any two edges of C intersected by is empty. If a prismatic k-

         circuit meets the edges

         e1, e2 ,…, ek

         of C successvely then we say that the edges

         F1, F2 ,…, Fk

         are an k-prismatic element of C.

         simplex and suppose that non-obtuse angles 0 ij 2

         are given corresponding to each

         edge

         Fij Fi Fj

         of C where Fi

         and Fj

         are the faces of C. Then there exist a compact

         hyperbolic polyhedron P in 3-dimensional hyperbolic space which realize C with dihedral

         angles

         ij

         at the edge

         Fij

         if and only if the following five conditions hold:

         1. C is trivalent.

         2. If

            Fijk Fi Fj Fk

            is a vertex of C then

            ij jk ki .

         3. If is a prismatic 3-circuit which intersects edges

            ij jk ki

            Fij , Fjk , Fki

            of C then

         4. If is a prismatic 4-circuit which intersects edges

            ij jk kl li 2 .

            Fij , Fjk , Fki , Fli

            of C then

         5. If Fs

         is a four sides faces of C with edges

         Fis , Fjs , Fks , Fls

         enumerated

         successively, then

         is ks ij jk kl li 3

         js ls ij jk kl li 3

         Furthermore, this polyhedron is unique up to hyperbolic isometries. Also Roeder, Hubbard and Dunbar proved that if C is not a triangular prism, then condition (5) is a consequence of

         (3) and (4) (Sec [14]). Andreevs restriction to non-obtuse dihedral angles is necessary to ensure that P be convex. Without this restriction of dihedral angles, compact hyperbolic polyhedral realizing a given abstract polyhedron may not be convex. Since dihedral angles of Coxeter polyhedron is non-obtuse, Andreevs theorem provide a complete characterization of 3-dimensional hyperbolic Coxeter polyhedron having more than four faces.

   2. Definitation 2.2.1. We denote k(P) the dimension of the projective group acting on a convex polyhedron P.

      3 if P is a tetrahedron,

      1 if P is a cone with base

      k(P)

      a convex polygon which is no

      0 otherwise

      Definition 2.2.2. A Coxeter group is an abstract group define by a group presentation of form

      nij

      nij

      .

      .

      Ri, ;Ri Rj ,i, j I

      Where I is a countable index set, nij N is symmetric for i, j and nij 1 .

      The fundamental group of the orbifold will be a Coxeter group with a presentation

      i i j

      i i j

      R ,i 1, 2,… f , R R nij 1

      where

      Ri is associated with silvered sides and

      Ri, j

      has order

      ni, j associated with the edge

      formed by the i-th and j-th side meeting.

      A Coxeter orbifold whose polytope has a side F and a vertex v where all other sides are adjacent triangles to F and contains v and all edge orders of F are 2 is called a cone-type Coxeter orbifold. A Coxeter orbifold whose polyhedron is topologically a polygon times an interval and edges orders of top and bottom sides are 2 is called a product-type Coxeter

      orbifold. If P

      is not above type and has not finite fundamental group, then P

      is said to be a

      normal-type Coxeter orbifold.

      be given a normal type

      Coxeter orbifold structure. Let k(P) be the dimension of the group of projective

      automorphisms acting on P. Suppose that P

      is orderable. Then the restricted deformation

      space of projective structures on the orbifold P

      is a smooth manifold of dimension

   1. f e e2 k P if it is not empty.

   P be given a normal type Coxeter

   orbifold structure. If space is empty.

   3 f e e2 0

   and P

   is orderable, then the restricted deformation

   Remark 2.2.5. Let P be a convex polyhedron and

   P be given a normal type Coxeter orbifold

   structure. Let k(P) be the dimension of the group of projective automorphisms acting on P.

   Suppose that P is orderable. Then P

   is projectively deformable if and only if

   3 f e e2 k P 0

   Definition 2.3.1. A planar graph is a graph that can be drawn on the sphere( or the plane) without edge crossings. Two edges of a graph are parallel if they have the same endpoints. A loop is an edge whose endpoints are the same vertex. If there are neither parallel edges nor loops, a graph is called simple. A simple graph is called k-connected if the removal of any k-1 or fewer vertices (all the edges they are incident with) leaves a connected graph. The dual graph of a plane graph is a plane graph obtained from the original graph by exchanging the vertices and faces. The dual graph of a graph is k-connected if and only if the graph is k- connected. If all the faces of planar graph is triangles then the graph is called triangulation. The dual of a triangulation is a trivalent planar graph. A triangulation with n vertices has

   exactly 3n 6 edges and 2n 4

   faces.

   G1 V1, E1 and G2 V2 , E2

   be two graphs imbedded on the sphere

   such that

   V1,V2

   be the set of vertices of G1,G2

   and

   E1, E2

   be the sets of edges of G1,G2 . An

   isomorphism from G1

   to G2

   is a pair of bijections

   :V1 V2

   and

   : E1 E2

   which

   preserve the vertex-edge incidence relationship.

   Definition 2.3.3. Let P be a convex polyhedron. The vertices and the edges of P from an abstract, finite, simple graph, called the graph of P and denoted by G(P). Thus, G(P) is an abstract graph defined on the set of vertices vert (P) of P. Two vertices u and v in vert (P) are adjacent in G(P) if and only if [u,v] is an edge of P.

   Definition 2.3.4. A 3-dimensional polyhedron is called simplicial polyhedron if every face contain exactly 3 vertices. A 3-dimensional polyhedron is called a simple polyhedron if each vertex is the intersection of exactly 3 faces.

   Theorem 2.3.5 (Blind and Mani). If P is convex polyhedron, then the graph G(P) determines the entire combinatorial structure of P.

   In other words, if two simple polyhedral have isomorphic graphs, then their combinatorial polyhedral are isomorphic as well.

   Steinitz established the following basic theory for 3-dimensional polyhedron.

   Since the compact hyperbolic polyhedron is simple, the combinatorial polyhedron of a compact hyperbolic polyhedron can be known from 3-connected planar graph of the polyhedron.

3. 1. Let P be a CH-Coxeter polyhedron and

      P be its Coxeter orbifold structure of P.

      P is orderable, then P is also orderable.

      Remark 3.1.2. If P is not orderable, then

      P is also not orderable.

      Let e be the number of edges of P and e2

      be the number of edges of edge order 2.

      Let v be the number of vertices of P and f be the number of faces of P.

      be Coxeter orbifold structure of P. Suppose P

      Then

      is orderable and projectively deformable.

      1. Every vertex is incident with exactly three edges.

      2. Every vertex is incident with at least one edge of edge order 2.

      3. v e

         5 v 10 .

         2 2

      4. v is even.

   3.2 Main Results. Now we are ready to establish the main results.

   be its Coxeter orbifold structure of P. If P

   e2 5 .

   is orderable and projectively deformable then

   Proof. By Chois theorem 2.2.3, 3 f e e2 k P 0 . By definition 2.2.1, for the polyhedra with 10 vertices P, k P 0 .

   Since the combinatorial structure of orderable and projectively deformable compact

   hyperbolic coxeter polyhedra with 10 vertices are

   P10-1, P10-2 , therefore

   f 7, e 15 .

   Since f

   7, e 15 , therefore

   3 f e e2 k T 0 3 7 15 e2 0 0 6 e2

   By proposition 3.1.3, every vertex is incident with at least one edge of edge order 2. Since there are 10 vertices of T and each vertex incident with exactly two vertices, therefore the

   number of edges of order 2 is at least v 5 . Hence e 5 .

   2 2

   Since 5 e2 6 , therefore e2 5 .

   be its Coxeter orbifold structure of P. If P is orderable and projectively deformable then the

   order of the edges at each vertex is one of the form 2,3,3,2,3, 4, 2,3,5 .

   Proof. Since there are 5 edges of P of order 2 and 10 vertices in P and each vertex is incident with at least one edge of order 2, therefore each vertex is incident with exactly one edge of order 2.

   Suppose that r1, r2 be the order of the two edges at a vertex other than 2. Then r1, r2 3 . By Andreevs first condition, we have

   1 1 1 1

   2 r1 r2

   1 1 1

   r1 r2 2

   1 2

   1 2

   Since r , r 3 , therefore

   1 1 2 . Hence

   1 1 1 2 .

   r1 r2 3 2 r1 r2 3

   1 2

   1 2

   1 2

   1 2

   If r , r 4 then 1 1 1 . This is a contradiction. Hence r 3 or r

   3.

   r1 r2 2

   Assume that

   r2 3 . If r1 6 then

   1 1 1 1 1 1 1 1 1 2 1 r 3 r 2

   2 r r 2 6 r 2 6 r 6 r 2 2

   1 2 2 2 2

   This is a contradiction. Hence 3 r1 5 .

   Therefore the edge order at each vertex is one of the form 2,3,3,2,3, 4, 2,3,5 .

   P10-1

   be its Coxeter orbifold structure. Suppose that

   P10-1

   is orderable and projectively

   deformable. Then the total number of such P10-1 is 5 and P10-1 is one of the tetrahedrons in figure 2.

   Proof. Suppose that the order of the edges be follows:

   r1, r2 ,…, r15

   and level the faces of P10-1 as

   From the above figure-3, the 3-prismetric circuits are F2 , F5 , F3 ,F4 , F7 , F3 ,F4 , F5 , F3 . Therefore by Andreevs second condition, we have

   1 1 1

   1, 1 1 1

   1, 1 1 1 1

   (1)

   r1 r6

   r11

   r3 r7

   r11

   r3 r12

   r15

   Therefore only one edge can be order of 2 in each set ofr1, r6 , r11,r3 , r7 , r11,r3 , r12 , r15

   and

   2, 3cannot be a subset of each set. Since there are five edges of order 2 and each vertex is incident with an edge of order 2, therefore all the edges of order 2 are disjoint.

   By definition of orderability 1.0.4, the order of the faces depends on the edges of order 2 and

   faces of higher index. Therefore we first assign edge order 2.

   Suppose that

   r1 2 . Then

   r6 , r11 2

   by equation (1). Since all the edges of order 2 are

   disjoint, therefore

   r5 , r4 , r3 , r2 2

   and hence

   r10 r7 2 . Since

   r7 2 , therefore

   r13 , r12 2 .

   Since at the vertex

   v10 , one edge is of order 2, therefore

   r15 2

   and hence

   r8 2 . Since five

   edges are order 2, therefore remaining edges must have order other than 2. In this case the

   order of the faces are

   F1, F6 , F2 , F7 , F4 , F5 , F3 .

   Suppose that r5 2 . Since all the edges of order 2 are disjoint, therefore r1, r4 , r6 , r10 2 .

   Since at the vertex

   v5 , one edge is of order 2, therefore

   r11 2 . From the equation (1), we

   have

   r3 , r7 , r1, r6 2 . Therefore

   r2 2 . Since

   r11 2

   at vertex

   v10 , therefore

   r13 , r15 2 . Then

   r12 2

   and

   r9 2 . Since five edges are order 2, therefore remaining edges must have order

   other than 2. Since 2, 3 cannot be a subset of each setr1, r6 , r11,r3 , r7 , r11,r3 , r12 , r15and

   2r1, r6 , r11r3 , r12 , r15, therefore

   3r1, r6 , r11r3 , r12 , r15. Thus

   r1 3, r3 3 . But the

   edge order at each vertex is one of the form 2,3,3,2,3, 4, 2,3,5

   by proposition 3.2.2. At

   the vertex v1 , the edge order r2 2 , therefore one of r1, r3 must be 3. This is a contradiction.

   Suppose that r4 2 . By symmetry r4 r9 , we will arrive similar situation as the case r5 2 .

   Therefore there is only one polyhedral P10-1 with edge order 2 which is as follows:

   Since 3r1, r6 , r11r3 , r12 , r15, therefore

   r3 3, r6 3, r11 3. Since at each vertex there is at

   least one edge of order 3, therefore r2 r4 r5 r9 r13 3 . Since r12 3 , therefore r14 3 .

   From the equations (1), we have

   1 1

   1 , 1 1

   1 , 1 1 1

   r6 r11

   2 r3 r11

   2 r3 r12 2

   Therefore 4 r6 , r11, r3 , r11, r12 5 and r6 , r11 ,r3 , r11 ,r3 , r12 are not identical with 4, 4 .

   Suppose that r3 4 . Then r11 r12 5 . We can choose r6 as 4 or 5.

   Suppose that

   r3 5 . If

   r11 4

   then we will get same polyhedron by symmetry

   r3

   r11 .

   Therefore we choose r11 5 . By symmetry r12 r6 , r12 4, r6 5 and r12 5, r6 4 will give

   same polyhedron up to symmetry. Therefore we can choose r12 , r6 as 4, 4,4,5,5,5 . Therefore we have exactly five orderable and deformable compact hyperbolic coxeter

   polyhedra P10-1 and these are as in figure-2. The order of the faces is

   F1, F6 , F2 , F7 , F4 , F5 , F3

   for each of the above polyhedrons. Therefore each of these above polyhedrons has ODCH Coxeter Orbifold structure.

In this article, we find that the number of orderable and deformable compact hyperbolic Coxeter polyhedral P10-1 in real projective space is exactly five. It can be extended to find all the 3-dimensional compact hyperbolic Coxeter polyhedral which are orderable and deformable in real projective space.

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Corresponding Author: Dhrubajit Choudhury, The Director, HR Foundation, H-19, Shankar Path, Hatigaon, Guwahati, Assam, India-781038