CLASSIFICATION OF ORDERABLE AND DEFORMABLE COMPACT COXETER POLYHEDRA IN HYPERBOLIC SPACE

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1 CLASSIFICATION OF ORDERABLE AND DEFORMABLE COMPACT COXETER POLYHEDRA IN HYPERBOLIC SPACE DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Abstract. The aim of this work is to investigate properties of 3-dimensional compact hyperbolic Coxeter polytope which are orderable and deformable in real projective space. We give a complete classification, up to isometry, of all 3-dimensional compact hyperbolic Coxeter polytope which are orderable and projectively deformable. By Andreev s and Choi s theorem, we proved that number of vertices of polyhedra in such family can t be more than 10. Using plantri program, we get the complete list of polyhedra having not more than 10 vertices. Using computer program, from these polyhedra, we found all the 3-dimensional compact hyperbolic Coxeter polytope which are orderable and deformable in real projective space. We also explain the algorithms of computer program behind this result. Introduction A n-dimensional orbifold is a Hausdorff space with a structure based on the quotient space of R n by a finite group action. We only deal with an orbifold of which universal cover is a manifold, the so-called a good orbifold. To give a hyperbolic structure on an orbifold, we model it locally by the orbit spaces of finite subgroups of P O(1, n) acting on open subsets of H n. Similarly, utilizing P GL(n + 1, R) and RP n we can give a real projective structure on an orbifold. A real projective structure on an orbifold M implies that M has a universal cover M and the deck transformation group π 1 (M) acting on M so that M/π 1 (M) is homeomorphic to M. Given a real projective structure on an orbifold M, we can define an immersion D from the universal cover M to RP n and a homomorphism h : π 1 (M) P GL(n + 1, R), for the orbifold fundamental group π 1 (M) of M, where D is called a developing map and h a holonomy homomorphism. However, (D, h) is determined only up to the following action (D, h( )) (g D, g h( ) g 1 ). for g P GL(n + 1, R). Conversely, the development pair (D, h) determines the real projective structure. Define D(M) by the space of equivalence classes of development pairs of real projective structures on M under the Date: October 07, 2009 and, in revised form, July,. Key words and phrases. Coxeter polyhedron, Orderable Coxeter polyhedron. The first author was supported in part by NSF Grant # This work was completed with the support of an BK 21 Scholarship. 1

2 2 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE isotopy action of M commuting with deck-transformation group. Here, the space of development pairs is equipped with C 1 -topology and D(M) is endowed with the quotient topology. The deformation space D(M) of real projective structure on the orbifold M is obtained from D(M) as a quotient space by the above action of P GL(n + 1, R) (See [3], [5]). The complement of a hyperspace in RP n can be identified with an affine space R n, the so-called an affine patch. A convex set in RP n is a convex set in an affine patch. If we use Klein s model of a n-dimensional hyperbolic space H n, then H n is an open ball in RP n and P O(1, n) is a subgroup of P GL(n + 1, R) preserving H n. Furthermore, H n can be imbedded in an (n + 1)-dimensional real vector space V as an upper part of hyperboloid x x x 2 n+1 = 1. Hence hyperbolic orbifolds or manifolds naturally have real projective structures. 1. Andreev s Theorem In 1970, E. M. Andreev provides a complete characterization of 3-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles on his paper On Convex Polyhedra in Lobacevskii Space. Therefore Andreev s theorem is a fundamental tool for classification of 3-dimensional compact hyperbolic Coxeter polytope. Some elementary facts about polyhedra are essential before we state Andreev s theorem. Let X be S 3, E 3, or H 3. Let Isom(X) denotes the group of isometries of X. A Coxeter polytope is a convex polytope whose dihedral angles are all submultiple of π. Let P be a 3-dimensional Coxeter polytope and Γ be the group generated by the reflections in the faces of P. Then Γ is a discrete group of Isom(X) and P is its fundamental polyhedron. Conversely, every discrete group Γ of Isom(X) can be obtained from a Coxeter polytope P such that P is its fundamental 3-dimensional polytope. Topologically, a 3-dimensional compact polytope P is a 3-dimensional ball and its boundary P is a 2-sphere S 2. The face structure of P gives a cell complex C of S 2 whose faces correspond to the faces of P, and so on. A polytope is called trivalent if degree of each vertex is 3. Roeder and Hubbard proved the following theorem considering compact hyperbolic polyhedra having non-obtuse dihedral angles. Theorem Suppose P be a compact hyperbolic polyhedron with nonobtuse dihedral angles. Then (a)there are exactly three edges incident at each vertex of P. i.e degree of each vertex of P is 3. (b)for such a P, angle of the faces are also 1 2 π Definition A cell complex on S 2 is called trivalent if each vertex is the intersection of three faces. A 3-dimensional abstract polytope is a trivalent cell complex C on S 2 that satisfied the following condition: (1) Every edge of C is the intersection of exactly two faces. (2) A non-empty intersection of two faces is either an edge or a vertex.

3 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 3 (3) Every face contains not fewer than three vertices or edges. If a face contains n vertices then n is called the length of the face Suppose C be the dual complex of C in S 2. Then C is a simplicial complex which embed in the same S 2 so that the vertex correspond to face of C, etc. A simple closed curve Γ in C is called k-circuit if it is formed by k edges of C. A k-circuit Γ is called prismatic k-circuit if the intersection of any two edges of C intersected by Γ is empty. If a prismatic k-circuit meets the edges e 1, e 2,..., e k of C successively then we say that the edges F 1, F 2,..., F k are an k-prismatic element of C. Theorem If Γ is a 3-circuit that is not prismatic in an abstract polyhedron C intersecting edges e 1, e 2, e 3, then edges e 1, e 2, and e 3 meet at a vertex. Proof. Suppose Γ = {F 1, F 2, F 3 } where F 1 F 2 = e 1, F 2 F 3 = e 2 and F 3 F 1 = e 3. Since Γ is a 3-circuit which is not prismatic, a pair of edges meet at a vertex. Suppose e 1 and e 2 meet at a vertex v 1. Then v 1 is a vertex of F 3 F 1. But F 3 F 1 = e 3. Therefore v 1 is a vertex of e 3. Hence e 1, e 2 and e 3 meet at v 1. Theorem (Andreev s Theorem). Let C be an abstract polyhedron such that C is not a simplex and suppose that non-obtuse angles α ij are given corresponding to each edge F ij = F i F j of C where F i and F j are the faces of C. Then there exist a compact hyperbolic polyhedron P in 3- dimensional hyperbolic space which realize C with dihedral angle α ij at the edge F ij if and only if the following five conditions hold: (1) 0 < α π 2. (2) If F ijk = F i F j F k is a vertex of C then α ij + α jk + α ki > π. (3) If Γ is a prismatic 3-circuit which intersects edges F ij, F jk, F ki of C then α ij + α jk + α ki < π. (4) If Γ is a prismatic 4-circuit which intersects edges F ij, F jk, F kl, F li of C then α ij + α jk + α kl + α li < 2π. (5) If F s is a four sided face of C with edges F is, F js, F ks, F ls enumerated successively, then α is + α ks + α ij + α jk + α kl + α li < 3π α js + α ls + α ij + α jk + α kl + α li < 3π Furthermore, this polyhedron is unique up to hyperbolic isometries. Also if C is not a triangular prism, then condition (5) is a consequence of (3) and (4). Andreev s restriction to non-obtuse dihedral angles is necessary to ensure that P be convex. Without this restriction of dihedral angles, compact hyperbolic polyhedra realizing a given abstract polyhedron may not be convex. Since dihedral angles of Coxeter polytope is non-obtuse, Andreev s theorem provide a complete characterization of 3-dimensional hyperbolic Coxeter polytope having more than four faces.

4 4 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE 2. Choi s Result from the article The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds 2.1. Introduction. In the article The deformation space of projective structure on 3-dimensinal Coxeter orbifolds, Prof. Choi stated a necessary and sufficient condition for 3-dimensional orderable and projectively deformable Coxeter polytopes. In this article, Prof. Choi concentrated on 3-dimensional orbifolds whose base spaces are homeomorphic to a convex polyhedron and whose sides are silvered and each edge is given an order. For example, a hyperbolic polyhedron with edge angles of form π/m for positive integer m will have a natural orbifold structure. If edge angle is π/m then edge order of this edge is m. Definition A Coxeter group Γ is an abstract group defined by a group presentation of form (R i ; (R i R j ) n ij ), i, j I. where I is a countable index set, n ij N is symmetric for i, j and n ij=1. The fundamental group of the orbifold will be a Coxeter group with a presentation R i, i = 1, 2,...f, (R i R j ) n ij = 1. where R i is associated with silvered sides and R i R j has order n ij associated with the edge formed by the i-th and j-th side meeting. Let P be a fixed convex polyhedron. Let us assign orders at each edge. Let e be the number of edges and e 2 be the numbers of order-two. Let f be the 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), i.e., 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 orbif old structure. 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 is whose polytope is topologically a polygon times an interval and edges orders of top and bottom sides are 2 is called 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. Definition An isotopy of an orbifold M is an orbifold-homomorphism f : M M such that there exists an orbifold map H : M I : M which restricts to an identity for t = 0 and restricts to f for t = 1. Definition The deformation space D( P ) of projective structures on an orbifold P is the space of all projective structures on P quotient by isotopy group actions of P.

5 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 5 A point p of D( P ) gives a fundamental polyhedron P upto projective automorphisms. The space where the fundamental polyhedron is always projectively equivalent to P is called the restricted def ormation space and denoted by D P ( P ). A point p in D P ( P ) is said to be hyperbolic if it is given by a hyperbolic structure on P. Suppose that p is a hyperbolic point of D P ( P ). We call a neighborhood of p in D P ( P ) the actual deformation space of P. We say that P is deformable, or it deforms into a real projective structure if the dimension of its actual deformation space is positive. Conversely, we say that P is projectively rigid, or simply rigid if the dimension of its actual deformation space is Orderable 3-dimensional potytope. We will consider a class of 3- dimensional potytope which is called orderable. In our study, We only concentrate on this class of 3-dimensional polytope. Definition We say P is orderable if we can order the sides of P so that each side meets sides of higher index in less than or equal to 3 edges. Example Each triangular side of P is said to be a class 0 side of P. A side of P meeting three or less number of sides which are not of class 0 be said to have class 1. Inductively, a side of P meeting three or less number of sides which are not of class i will be said to have class i + 1. P satisfiea trivalent condition if each side has a finite class order. In this case, we can order the sides by ordering from class 0 and then to the next class and so on; P is orderable. Remark Cube and dodecahedron are not satisfying orderability condition. Definition We denote k(p ) the dimension of the projective group acting on a convex polyhedron P. This dimension k(p ) of the subgroup of G acting on P equals 3 if P is a tetrahedron and equals 1 if P is a cone with base a convex polygon which is not a triangle. Otherwise, k(p ) = 0. Definition Let P be the orbifold structure of a 3-dimensional polytope 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. In the article[7], Prof. Choi proved the following theorem: Theorem 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 the restricted deformation space of projective structures on the orbifold P is a smooth manifold of dimension 3f e e 2 k(p ) if it is not empty. Corollary Let P be a convex polyhedron and P be given a normal type Coxeter orbifold structure. If 3f e e 2 < 0 and P is orderable, then the restricted deformation space is empty.

6 6 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Remark 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 3f e e 2 k(p ) > Main Results 3.1. Properties of Compact hyperbolic Coxeter Polyhedra. Suppose that P is a 3-dimensional Coxeter polytope in hyperbolic space. Let e be the number of edges of P and e 2 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. Theorem Let P be a 3-dimensional compact hyperbolic Coxeter polytope and P be Coxeter orbifold structure of P. Suppose P is orderable and projectively deformable. Then 1. Every vertex is incident with exactly three edges. 2. Every vertex is incident with at least one edge of edge order v 2 e v is even. Proof. 1.Since P is a 3-dimensional compact hyperbolic polytope. By theorem 1.0.1(a), degree of each vertex of P is 3 i.e. every vertex is incident with exactly three edges. 2. Suppose e 1, e 2, e 3 be the edge order of the edges incident at a vertex say v 1 of P. Since P is compact hyperbolic, therefore by Andreev theorem 1 e e e 3 > 1. If e 1, e 2, e 3 3 then 1 < 1 e e e 3 < = 1. This is not possible. Therefore at least one of e 1, e 2, e 3 must be edge order two. Hence edge order of at least one edge is 2 at each vertex. 3. We know that every edge is adjacent with exactly two vertices and therefore all e 2 edges of order 2 are incident with at most 2e 2 vertices. Again every vertex is incident with at least one edge of order 2. Therefore v 2e 2 and hence e 2 v 2. Suppose P is orderable and deformable. Since P is 3-dimensional compact hyperbolic polytope, therefore degree of each vertex is 3 and hence e = 3v/2. By Euler equation, v e + f = 2 i.e., f = 3v/2 v + 2 = v/ By Choi s theorem, 3f e e 2 k(p ) > 0 3v/ v/2 e 2 k(p ) > 0 e 2 < 6 k(p ). i.e. e The vertex and edge relation 2e = 3v implies that 3v is even. Therefore v is even. Corollary The number of vertices of a 3-dimensional compact hyperbolic polytope is at most 10.

7 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 7 Proof. By the above theorem, v 2 e Hence v Every edge adjacent with exactly two vertices and every vertex incident with at least one edge of edge order two, therefore number of vertices of P can t be more than 10. Hence v 10. Theorem Let P be Coxeter orbifold structure of a 3-dimensional compact hyperbolic polytope P. If P is orderable then P also orderable. Proof. Suppose P is orderable. Assume that F 1, F 2, F 3,... be the face order of the faces of P. Then by definition of orderable of P, number of edges of face F i which are of edge order two or edges of faces F i+1, F i+2, F i+3, Therefore number of edges of face F i which are edges of face F i+1, F i+2, F i+3, By definition of orderable of P, F 1, F 2, F 3,... be the face order of the faces of P and hence P is orderable. Remark If P is not orderable then P is also not orderable. Theorem Let P be a 3-dimensional orderable compact hyperbolic polytope. Assume that F 1, F 2, F 3,... be the face order of the faces of P. Then number of edges in the face F i i.e., length of F i can t be more than i + 2. Proof. If possible, let length of F i > i + 2. Maximum number of edges of F i included in a side of lower index is i 1. Therefore minimum number of edges of F i included in a side of higher index is (i + 3) (i 1) = 4. Therefore length of F i i + 2 Corollary If P is a 3-dimensional orderable polytope then P has at least one triangular face. Proof. From above theorem, length of F 1 3. triangular face. Therefore F 1 must be a Corollary Let P be orderable 3-dimensional polytope. Suppose P has exactly one face of length i. Then P has at least one face of length i + 1. Theorem Let P be a 3-dimensional compact hyperbolic polytope. Let F 1 and F 2 be two distinct triangular faces of P. Then P is a tetrahedron if and only if F 1 F 2. Proof. If P is a tetrahedron then F i F j for any two distinct faces F i, F j of P. Conversely, let F 1 F 2. Since P is a 3-dimensional compact hyperbolic polytope, degree of each vertex is 3. Then F 1 F 2 is an edge. Let F 1 = {v 1, v 2, v 3 } and F 2 = {v 1, v 2, v 4 } up to labeling the vertices as in figure-1. Let F 3 and F 4 be the faces of P such that v 1 is a vertex of F 3 and v 2 is a vertex of F 4. Then v 3, v 4 F 3 F 4. Therefore {v 3, v 4 } is an edge of P. Then degree of each vertices v i for i = 1, 2, 3, 4 is 3 and hence v 1, v 2, v 3, v 4 are the only vertices of P. Therefore P is a tetrahedron. Theorem Let P be a 3-dimensional compact hyperbolic polytope. Let F 1, F 2 be two distinct triangular faces of P such that F 1 F 3 =. Let F 3

8 8 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE F 3 v 1 v 3 v 4 F1 F 2 v 2 F 4 Figure 1 is another face of P such that F 1 F 3 and F 2 F 3. Then either P is a prism or length of F 3 5. F 4 v 1 v 4 v 5 v 6 F1 F 3 F 2 v 2 v 3 F 5 Figure 2 Proof. Suppose that the length of F 3 is 4. Since we have F 1 F 2 =, P is not a tetrahedron. Since we have F 1 F 3 and F 2 F 3, F 1 F 3 and F 2 F 3 are edges of F 3. Then we can assume F 1 = {v 5, v 1, v 2 }, F 2 = {v 6, v 3, v 4 }, F 3 = {v 1, v 2, v 3, v 4 } up to labeling the vertices as in figure-2. Let F 4, F 5 be the faces of P such that {v 1, v 4 } is an edge of F 4 and {v 2, v 3 } is an edge of F 5.

9 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 9 Then F 1, F 3 and F 4 are the faces at the vertex v 1. Since P is a 3-dimensional compact hyperbolic polytope, degree of each vertex is 3. Therefore {v 1, v 5 } is an edge of F 4. Similarly, {v 4, v 6 } is an edge of F 4. In the same way, we can find that {v 2, v 5 }, {v 3, v 6 } are edges of F 5. Hence v 5, v 6 F 4 F 5 and F 4 F 5 is the edge {v 5, v 6 }. Therefore F 4 = {v 1, v 4, v 6, v 5 } and F 5 = {v 2, v 3, v 6, v 5 }. Hence P is a prism. Theorem Let P be a 3-dimensional polytope such that P has more than 8 vertices.i.e., v > 8. Suppose P be Coxeter orbifold structure of P. If P is orderable and deformable compact hyperbolic 3-dimensional polytope then edge orders of the three edges at each vertex is one of the following form (2, 3, 3), (2, 3, 4), (2, 3, 5) Proof. Since P is 3-dimensional ordrable and deformable compact hyperbolic polytope with v > 8. By theorem 2.0.1, v 10 and v is even number and hence v = 10. Again by theorem 2.0.1, v 2 e 2 5. Therefore e 2 = 5. v 8 v 7 v 10 v 9 v 3 v v 1 n v v 4 v 5 Figure 3 Suppose the edge {v 1, v 2 } is of edge order n 6 as in the above figure. Then the 4 edges adjacent with v 1 or v 2 are of edge order 2. Since we have e 2 = 5, there are exactly one edge of edge order 2 which can be incident with at most two of among the 4 vertices v 7, v 8, v 9 and v 10. Hence there is at least two vertices which are not adjacent with edges of edge order 2. This is a contradiction. Therefore edges order of the three edges at each vertex can t be of the form (2, 2, n), n 6 and hence one of the form (2, 3, 3), (2, 3, 4), (2, 3, 5) Theorem Let P be a compact 3-dimensional hyperbolic Coxeter polytope and P be Coxeter orbifold structure of P. Suppose P is not a tetrahedron and P is orderable. If P is projectively deformable then the number of triangular faces of P is not more than two.

10 10 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Proof. Suppose that P has more than two triangular faces. Since P is not a tetrahedron, therefore all the triangular faces of P are mutually disjoint. Then the number of vertices i.e. v 9. Since P is an orderable and projectively deformable 3-dimensional compact hyperbolic polytope, therefore number of vertices i.e. v 10. Since P is a compact hyperbolic polytope, therefore v is even and hence v = 10. By theorem 3.1.8, we have that any two triangular faces of P are disjoint if v > 4, Therefore P has exactly three triangular faces. Then there is exactly one vertex S disjoint from triangular faces. Suppose S is adjacent with two vertices say v 1, v 3 of a triangular face {v 1, v 3, v 4 }. Then S form a triangular face {S, v 1, v 3 } and hence the number of triangular face 4. This is a contradiction. Hence there are at most one edge from S to an triangular face of P. Since P has exactly three faces disjoint from S and all the faces disjoint from S are triangular, there is exactly one edge from S to each triangular face of P. Suppose there is two edges say {v 1, v 2 }, {v 4, v 5 } between two triangular faces {v 1, v 3, v 4 }, {v 2, v 5, v 6 }. Then {v 1, v 2, v 5, v 4 } is a face of length 4 and adjacent with two triangular faces. By theorem 3.1.9, P is a prism. This is a contradiction. Therefore there are at most one edge between two disjoint triangular faces. v 1 v 2 e 2 e 5 e 3 e 4 v 3 v 6 v e 6 e 7 4 S e 11 e 9 v 5 e 12 v 10 e 14 e 15 v 8 e v 9 13 Figure 4 Since total number of edges in P = 15 and there are 12 edges in the above figure 4. So we need to add exactly 3 edges in the above figure. Let take an edge e 1 at v 1. Then e 1 must be adjacent with one of the vertices v 2, v 5, v 8 and v 9. By symmetry of the figure 4, we can assume e 1 = {v 1, v 2 }. Now take the edge e 8 at vertex v 4. Then e 8 can t adjacent with v 5. Therefore e 8 must be adjacent with one of v 8, v 9. By symmetry, we can assume e 8 = {v 4, v 8 }. Remaining one edge must be e 10 = {v 5, v 10 }. Therefore P is of the form in figure 5.

11 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 11 v 1 e 1 v 2 e 2 e 5 e 3 e 4 v 3 v 6 v e 6 e 7 4 S e 11 e 9 v 5 e 12 v 10 e 8 e 10 e 14 e 15 v 8 e v 9 13 Figure 5 Let α i be the order of the edge e i. By the previous theorem, order of the edges at each vertex is one of the following form (2, 3, 3), (2, 3, 4), (2, 3, 5) Without loss of generality, we can assume α 7 = 2. Since {e 1, e 7, e 8 } is a 3-prismatic elements of P. Therefore 4 α 1, α 8 5. Then α 3 = 2, α 6 = α 2 = 3. One of α 4, α 5 is 2 and one of α 5, α 9 is 2. Also only one of α 4, α 5, α 9 is 2. Hence α 5 = 2, α 4 = 3. Since α 9 2, therefore α 10 = 2 and hence α 14 = 2. Since {e 1, e 11, e 10 } and {e 8, e 12, e 10 } are 3-prismatic elements of P. Therefore 4 α 11, α Since P is a compact hyperbolic Coxeter polytope. Therefore at the vertex v 7, by 1st condition of Andreev s theorem we get > 1 α 7 α 11 α 12 But α 7 α 11 α = 1 This contradicts our assumption. Therefore the number of triangular faces of P is not more than two. Theorem Let P be a Compact hyperbolic polytope and P be its Coxeter orbifold structure. Then (1) Any two distinct faces F 1, F 2 contains at least l 1 + l 2 2 distinct vertices of P, where l i is the lenth of F i. (2) Any three distinct faces F 1, F 2 and F 3 contains al least l 1 +l 2 +l 3 6 distinct vertices of P. Proof. Since P is a convex polytope, any two faces have at most one vertex or one edge common. So F 1 F 2 contains V (F 1 ) + V (F 2 ) V (F 1 F 2 ) vertices. Since V (F 1 F 2 ) 2, F 1 F 2 contains at least l 1 + l 2 2 vertices. Again, (F 1 F 2 F 3 ) = (F 1 )+ (F 1 )+ (F 1 ) (F 1 F 2 ) (F 1 F 3 ) (F 2

12 12 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE F 3 ) + (F 1 F 2 F 3 ). Therefore F 1 F 2 F 3 contains at least l 1 + l 2 + l 3 6 vertices. Theorem Let P be a compact hyperbolic polytope. Suppose P is orderable having more than 4 faces. Then P has exactly two triangular faces and both are disjoint. F 3 v 1 v 3 F1 F 2 F 4 v 2 Figure 6 Proof. From previous theorem, P can t have more than two triangular faces. Since P is orderable, P has at least one triangular face. Suppose that P has only one triangular face, say F 1. Suppose that adjacent faces of F 1 are F 2, F 3 and F 4. Case I: Suppose that v = 6. Then f = 5. Then there is one face say F 5 which is disjoint from F 1. F 5 doesn t contain v 1, v 2, v 3. Since v = 6, F 5 contains only remaining three vertices and hance F 5 is triangular face. This is contradiction. So v 6. Case II: Suppose v = 8. Then f = 6. There are two faces say F 5, F 6 disjoint from F 1 and contains 5 vertices. Therefore l 5 + l i.e, l 5 + l 6 7. So one of l 5, l 6 must be 3. Hence F 5 or F 6 is triangular face. This is contradiction. So v 8. Case III Suppose v = 10. Let F 1, F 2,... be satisfying orderability condition of the faces. Then F 1 is triangle and F 2 is of the length 4. Since P has only one triangular face, so F 2 is of length 4 and adjacent with F 1. Suppose that F 3, F 4 and F 5 are the faces adjacent with F 1 or F 2. Since v = 10, f = 7. Therefore there are two faces say F 6, F 7 disjoint from F 1 and F 2 and contains five vertices. Hence l 6 + l i.e l 6 + l 7 7. Therefore one of l 6, l 7 is 3. Hence one of F 6, F 7 is triangular. This is contradiction. Hence v 10. Hence there is no compact hyperbolic orderable polytope with one triangular face.

13 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 13 F 3 v 1 v 4 v 3 F1 F 2 F 5 v 2 F 4 Figure 7 v 5 Theorem Suppose P is 3-dimensional compact hyperbolic polytope. Suppose P is orderable. Then the total number of such polytopes is at most 5. Proof. Case I Suppose v = 4. Then P is tetrahedron. Case II Suppose v = 6. Then P has exactly two disjoint triangular faces. Suppose {v 1, v 2, v 3 }, {v 4, v 5, v 6 } be the triangular faces of P. Since degree of each vertex is 3, there is only one edge from each vertex of one triangle to another triangle. So abstract graph of P up to isometry is {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 4, v 5 }, {v 5, v 6 }, {v 1, v 6 }, {v 1, v 4 }, {v 2, v 5 }, {v 3, v 6 }}. This is a prism. Case III Suppose v = 8. Then P has two triangular faces say F 1 = {v 1, v 2, v 3 }, F 2 = {v 4, v 5, v 6 }. Let v 7, v 8 be the other two vertices of P. Suppose there are two edges from v 7 to two vertices v 1, v 2 of one triangular face F 1. Then {v 1, v 2, v 7 } is a triangular face. This contradicts the fact that P has two triangular faces. Hence from the vertex v 7 ( or v 8 ), there is at most one edge incident with vertices of triangular faces. Since degree of each vertex is 3, three edges from v 7 must be {v 7, v 1 }, {v 7, v 4 }, {v 7, v 8 } and the edges from v 8 must be {v 8, v 2 }, {v 8, v 5 }, {v 7, v 8 } up to isometry. Since {v 3, v 1 }, {v 3, v 2 } are two edges at v 3, another edge must be {v 3, v 4. Hence the abstract graph of P up to isometry is {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 4, v 5 }, {v 4, v 6 } {v 5, v 6 }, {v 7, v 1 }, {v 7, v 4 }, {v 7, v 8 }, {v 8, v 2 }, {v 8, v 5 }, {v 3, v 4 }}. Case IV Suppose v = 10. Then f = 7 and P has two triangular faces. Let F 1, F 2, F 3,... be the ordering of the faces of P. Then F 1 is triangle. Let F 1 = {v 1, v 2, v 3 }. Subcase 1: Suppose F 1 F 2. If F 2 is triangle then P must be tetrahedron. This is contradiction. Therefore the length of F 2 is

14 14 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE 4. Again by theorem 3.1.5, the length of F 2 i.e. l 2 is 4. Hence l 2 = 4. Let F 2 = {v 1, v 2, v 4, v 5 }. v 7 v 6 F6 F 7 v 10 v 8 v 9 F 3 v 1 v 4 v 3 F1 F 2 F 5 v 2 F 4 v 5 Figure 8 Suppose that F 3, F 4 and F 5 are the faces adjacent with F 1 or F 2. Since v = 10, f = 7. Therefore there are two faces say F 6, F 7 disjoint from F 1 and F 2 and contains five vertices. Hence l 6 + l i.e l 6 + l 7 7. Therefore one of l 6, l 7 is 3. Let l 6 = 3. Then l 7 = 4 and both F 6 and F 7 are adjacent. Let F 6 = {v 6, v 7, v 8 } and F 7 = {v 7, v 8, v 9, v 10 }. Now v 3 must adjacent with one of v 6, v 9 up to symmetry. (a) Suppose v 3 is adjacent with v 6. Then v 4 is adjacent with v 9 and v 5 is adjacent with v 10 up to symmetry. Therefore the graph of P is {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 1, v 4 }, {v 2, v 5 }, {v 4, v 5 },

15 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 15 {v 6, v 7 }, {v 7, v 8 }, {v 6, v 8 }, {v 7, v 10 }, {v 8, v 9 }, {v 9, v 10 }, {v 3, v 6 }, {v 4, v 9 }, {v 5, v 10 }} (b) Suppose v 3 is adjacent with v 9. Then v 4 is adjacent v 10 and v 5 is adjacent v 6 up to symmetry. Therefore the graph of P is {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 1, v 4 }, {v 2, v 5 }, {v 4, v 5 }, {v 6, v 7 }, {v 7, v 8 }, {v 6, v 8 }, {v 7, v 10 }, {v 8, v 9 }, {v 9, v 10 }, {v 3, v 9 }, {v 4, v 10 }, {v 5, v 6 } F 4 v 7 F 6 v 1 v 4 v 5 v F 6 1 F 3 F 2 v 2 v 3 F 5 Figure 9 Subcase 2: Suppose F 1 F 2 =. Then F 1 and F 2 are triangle. Since P has exactly two triangle,other faces are of length 4. If the length of F 3 is 4 then F 3 is adjacent with one of F 1 and F 2 say F 1. Clearly F 1, F 3, F 2,..., is also satisfied orderability condition and P is belongs to the subcase I with this new ordering of the faces. Assume the length of F 3 is 5. Then F 1 and F 2 are adjacent with F 3. From figure 9, only face F 7 is disjoint from F 1, F 2 and F 3. Also F 7 does not contain v 1,..., v 7. Hence F 7 = {v 8, v 9, v 10 }. Therefore P has three triangular faces which is a contradiction. Since P has exactly two triangular faces. Therefore total number of compact orderable polytope is at most 5.

16 16 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE 4. Plantri Program 4.1. Introduction: The program plantri is the fastest C program which generates certain type of graphs that are imbedded on the sphere or the plane. Exactly one graph of each isomorphism class is output. A 3-dimensional convex polytope can be represented by a 3-connected simple planner graph. Plantri program generates the list of 3-connected simple planer graphs of finite number of vertices and hence generates the list of 3-dimensional polytopes of finite number of vertices up to isomorphism. Definition 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 graph is called k-connected if there does not exist k 1 vertices whose removal makes the graph disconnected. The dual graph of a plane graph is a plane graph obtained from the original graph by exchanging the vertices and faces. 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. Definition Graph of a Polytope Let P be an abstract polytope. The graph G of P is an abstract graph defined on the vertices of P such that {u, v} is an edge in G if {u, v} is an edge of P. Theorem (Blind and Mani s Theorem ). If P is an abstract polytope then graph G of P determines the entire combinatorial structure of P. Theorem (Balinski s Theorem). The graph G of a 3-dimensional polytope is 3-connected simple planar graph. Theorem (Steinitz Theorem). Steinitz s proved the following relations between graph and polytope: (1) G is the graph of a 3-dimensional polytope if and only if it is simple, planar and 3-connected. (2) Every 3-connected planar graph has a representation in the plane such that all edges are straight, and all the bounded regions determined it, as well as the union of all the bounded regions, are convex polygons. (3) Every 3-connected planar graph is the graph of a 3-polytope P whose edges touch the unit sphere. Definition (Graph Isomorphism). Let G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) be two graphs imbedded on the sphere such that V 1, V 2 be the set of vertices of G 1, G 2 and E 1, E 2 be the sets of edges of G 1, G 2. An isomorphism from G 1 to G 2 is a pair of bijections φ : V 1 V 2 and ψ : E 1 E 2 which preserve the vertex-edge incidence relationship. Theorem (Schramm s Theorem). An isomorphism of a 3-connected planar graph is realized by a symmetry of convex polytope.i.e., An Isometry of two convex polytopes is an isomorphism of the 3-connected planar graphs of the convex polytopes.

17 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA Output Formats of Plantri program: Ascii code is a human readable version of planar code. The vertices of the the graph are named as Ascii characters starting with a. Each line of output represents a graph. Example The output format of a graph is 10 bcd, aef, agd, ach, bif, beg, cfj, dji, ehj, gih a b c f d g e j h i Figure 10 This is a graph with 10 vertices a, b, c, d, e, f, g, h, i, j. The neighboring vertices of a in clockwise order are b, c, d; and so on. Starting neighboring vertex of a vertex in output format is the lowest ascii characher. For example: b is the lowest ascii character among the neighbors of a and hence neighbors of a is started from b in clockwide direction. Installing plantri program: The latest edition of plantri can be obtained from http : //cs.anu.edu.au/ bdm/plantri 4.3. Algorithm of Computer Program. We are ready to give an outline of the entire computational procedure to classify 3-dimensional compact Coxeter polytopes which are orderable and projectively deformable. Based on the above theorems, we have the following properties of a 3-dimensional compact Coxeter polytope P : (1) 6 v 10 if P is not tetrahedron. (2) e 2 5 (3) Degree of each vertex is 3. Our aim of the program is to find edge order from 2 to 6 of the edges of the Coxeter polytope which is 3-dimensional orderable and projectively deformable compact hyperbolic polytope. Initial data is needed to start the program, i.e., 3-dimensional trivalent polytopes with v 10 can easily be

18 18 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE obtained using plantri program. Algorithm (1) Input the list of the 3-dimensional trivalent polytopes as 3-connected graphs such that 6 v 10 as the output format of plantri program. In the initial data file, each line is of the form: 10 bcd,aef,agd,ach,bif,beg,cfj,dji,ehj,gih, (2) Compute the program for each line of data file successively from top to bottom. (3) Labeling the vertices a, b, c,... by 1, 2, 3,... (4) Labeling the edges by e[1], e[2], e[3],... as follows: Let e[p] = {i 1, j 1 } i1 <j 1, e[q] = {i 2, j 2 } i2 <j 2. Then { if i1 < i p < q if 2, or if i 1 = i 2 andj 1 < j 2 (5) Find the faces of the polytope and labeling the faces by F [1], F [2], F [3],... as follows: (6) Find the dual of the graph. (7) Labeling the edges of the dual graph as follow: The edge {F [i], F [j]} is labeled by e[p] if F [i] F [j] = e[p]. (8) Find the 3-prismatic circuits and then 3-prismatic elements of the graph. (9) Find the 4-prismatic circuits and then 4-prismatic elements of the graph. (10) Choose the order of the edges from {2, 3, 4, 5, 6} such that e 2 5. (11) Check the Andreev s conditions (2), (3) and (4). (12) If the Andreev s conditions (2),(3),(4) are satisfied then check orderability condition. Output format of the computer program Suppose the input graph is 10 bcd,aef,agd,ach,bif,beg,cfj,dji,ehj,gih, Then output of the program is as follows: (1) Graph[index]: 234, 156, 174, 138, 296, 257, 3610, 4109, 5810, 798, where index is the numbering of the graphs in data file. Here 2, 3, 4 are the adjacent vertices of the vertex 1 in clockwise direction and so on. (2) (V, E, F ) = (10, 15, 7) where V = the number of the vertices, E = the number of the edges and F = the number of the faces. (3) Faces of the graph: , 1-3-4, , 2-5-6, , , ,; This are the faces F [1], F [2], F [3], F [4], F [5], F [6], F [7] successively.

19 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 19 In the face F [1] = , any two consecutive vertices of the face F [1] are adjacent and 1,3 are adjacent. i.e., vertices form a cycle. (4) Edges of the graph: (1, 2) = e[1], (1, 3) = e[2] and so on. Here e[1] is the edge between 1 and 2. (5) Edges of the dual graph: (1, 2) = e[2], (1, 3) = e[1] and so on. Here e[2] is the edge between the faces F [1] and F [2] and so on. (6) The number of 3-prismatic circuits: 3. (7) 3-prismatic curcuits are : 1,3,6; 1,3,5; 3,5,6; i.e., {F [1], F [3], F [6]}, {F [1], F [3], F [5]}, {F [3], F [5], F [6]} are the 3-prismatic circuits. (8) 3-prismatic elements: 1,10,11; 1,8,7; 8,12,10; i.e., the set of edges {e[1], e[10], e[11]}, {e[1], e[8], e[7]}, {e[8], e[12], e[10]} form 3-prismatic elements of the graph. (9) Graph[index] is orderable and projectively deformable compact hyperbolic Coxeter polytope with Edges order: and face order: 2, 1, 6, 3, 7, 4, 5, i.e., The order of the edges e[1], e[2], e[3],.. are 2, 3, 4,.. so on and F [2], F [1], F [6], F [3],.. satisfy the orderability condition of the faces. With these orderability of edges and faces, the graph is orderable and projectively deformable compact hyperbolic polytope. 5. Computer Results v 3 e 2 e 6 e 4 v 4 e 3 e 5 v 1 v e 2 1 Figure 11. Levels of the edges of tetrahedron

20 20 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 T T T T T Table 1. The list of Coxeter tetrahedra(the results are obtained from the book Foundations of Hyperbolic Manifolds. Page-296) F 3 v 1 e 1 v 2 e 2 e 5 F 1 e 3 F 2 v 3 v 6 F 4 e e 4 7 F 5 e 6 e 9 v 4 e v 5 8 Figure 12. Prism Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 P P P P P P P P P P P P P P P P P P Continued on next page

21 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 21 Table 2 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

22 22 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Table 2 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

23 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 23 Table 2 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 P P P P P P P P P P P Table 2: The list of Coxeter Prism s)

24 24 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE v 1 F 3 e 3 e 1 e 2 v 4 F 2 F 1 v 2 e 8 F e 7 v 3 e 6 e 5 6 v 7 v 6 F 4 e 9 e 4 F 5 e 12 e 10 e 11 v 8 v 5 Figure 13. Levels of the edges of III [h] Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P Continued on next page

25 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 25 Table 3 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

26 26 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Table 3 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

27 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 27 Table 3 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

28 28 DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Table 3 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page

29 CLASSIFICATION OF DEFORMABLE COMPACT COXETER POLYHEDRA 29 Table 3 continued from previous page Order of the edges Name e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P Continued on next page