1 Introduction. For example, consider four points A, B, C, D which form a convex polygon in the plane.

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1 TOPCOM: Triangulations of Point Configurations and Oriented Matroids - Jörg Rambau - Handout by Clemens Pohle - Student Seminar in Combinatorics: Mathematical Software 1 Introduction TOPCOM is a package written by Jörg Rambau which can be used to compute the number of triangulations of a given point configuration. But it provides also much more functionality, for example it s possible to compute the chirotope of a point configuration or the flip graph component of a triangulation. Before we continue, we give the definition of a triangulation. For a d-dimensional configuration A R d of n points we call a sub-configuration σ A a k-simplex if it consists of k + 1 affinely independent points. For example, a 1-simplex consists of two different points, a 2-simplex is a non-degenerate triangle and a 3-simplex is a (not necessarily regular) tetrahedron. We call a collection T of d-simplices a triangulation if the convex hulls of the d-simplices cover the convex hull of A and if the d-simplices intersect properly, i.e. σ, τ T : conv(σ τ) = conv(σ) conv(τ). For example, consider four points A, B, C, D which form a convex polygon in the plane. Then {{A, B, C}, {A, C, D}} is a triangulation, but {{A, B, C}, {A, C, D}, {B, C, D}} is not since conv({a, B, C} {B, C, D}) conv({a, B, C}) conv({b, C, D}). 1

2 Figure 1: k-simplices for k {1, 2, 3} 2 The Chirotope of a Point Configuration ( The chirotope of a point configuration A is a function χ : A d+1) {+,, 0} which assigns every (d+1)-subset of A its orientation. If the d+1 points of such a subset σ are not affinely independent, then χ(σ) = 0. If the points of A are given in homogeneous coordinates, then χ can be computed as χ({a 1,..., a d+1 }) = sign(det(a 1,..., a d+1 )). An example for the chirotope of a point configuration can be found in figure 2. Chirotope χ: Figure 2: Four points in the plane and their chirotope The interesting and useful fact about chirotopes is that they contain all the necessary information to compute the triangulations of a given point configuration. Further, by using the chirotope, round-off errors can be avoided. We ll discuss the commands of TOPCOM with a concrete example. For that purpose, consider the following point configuration in the plane: A 0 = {(0, 0), (1, 1), (3, 1), (5, 0), (1, 5)} 2

3 The point configuration can be seen in figure 3. Figure 3: The 2-dimensional point configuration A 0 To use A 0 in TOPCOM, we have to save [[0,0,1],[1,1,1],[3,1,1],[5,0,1][1,5,1]] as a txt-file. Let s say we called this file example.txt. Note that we transformed the coordinates into homogeneous coordinates. The command points2chiro computes the chirotope of a given point configuration. To use it, type points2chiro < example.txt into the terminal. As output, we get 5,3: is the number of points, 3 is equal to d + 1 and the sequence of plus and minus signs corresponds to the function values of χ of the (d + 1)-subsets of A 0, in lexicographic order. So for example, the first minus indicates that the orientation of the triangle 012 = {(0, 0), (1, 1), (3, 1)} is negative. 3 Placing Triangulations and Flips The following algorithm describes a very easy way to obtain a triangulation for a given configuration of points. We will call such a triangulation a placing triangulation. In each step k, we maintain the following sets: 3

4 A k : set of points that is already triangulated T k : placing triangulation of A k F k : set of all boundary facets of T k that are interior in A We take the first (d + 1)-subset S in lexicographic order which is a d-simplex and set A d+1 = S, T d+1 = {S} and F d+1 = {F S F is a facet of S and not in the boundary of A}. In each step, we add a point a k+1 to A k and all simplices F a k+1 to T k for which F is in F k and visible from a k+1. This yields A k+1 and T k+1. Here, visible means that χ(f a k+1 ) χ(s ) for all S T k with F S. Then, the visible facets are removed from F k and the new non-boundary facets of the new simplices are added to F k, yielding F k+1. The process stops when F k is empty for some k. Since we add a point in each step, the process stops at the latest when k = n. It s also possible that the process stops earlier. In that case, we didn t use all of the points for our triangulation. To compute a placing triangulation, one can use the command points2placingtriang. So in our example, we have to write into the terminal. As output, we get points2placingtriang < example.txt {{0,1,2},{0,2,3},{1,2,4},{0,1,4},{2,3,4}} We see that this triangulation makes sense. First, the algorithm starts with the simplex 012. Then, the point 3 is added, but the only facet visible from 3 is the facet 02, so we only add the triangle 023. In the last step, the point 4 is added, and the triangles 014, 124 and 234 since the facets 01, 12 and 23 are visible from 4. If we change the order of the points, then we might get another triangulation. For example, let s exchange point 1 and point 4 in our example. Then the new triangulation is {{0, 1, 2}, {0, 2, 3}, {1, 2, 3}}, as shown in figure 4. Note that this triangulation does not use the point 4. If one wants a triangulation which uses all the points, then one can flip in the missing points. A flip is the exchange between the two possible triangulations in a subset of d + 2 points. The possible flips for the cases d = 1 and d = 2 are shown in figure 5. 4

5 Figure 4: It s possible that not all points are used in a placing triangulation 4 Flip graph Figure 5: The possible flips for d = 1 and d = 2 Let A be a point configuration and denote by V the set of all triangulations of A. Then we can define a graph G which has the set V as vertices and where two vertices are connected by an edge if the corresponding triangulations only differ by a single flip. TOPCOM uses a Breadth-First-Search procedure to explore the component of a flip graph and constructs the edges dynamically during the exploration. To find the edges of a placing triangulation, one can use the command points2flips. In our example, points2flips < example.txt outputs the following: [5,3:[[1,3,0,2]->0,[0,2,4,1]->0]] This means that we can either replace the triangles 012 and 023 by 013 and 123 or flip out the point 1 in the triangle 024. Using points2triangs, one can get all triangulations in the flip graph component of the corresponding placing triangulation. In our 5

6 example, there are five triangulations, and the flip graph can be seen in figure 6. Figure 6: The flip graph of our example If one wants only the triangulations which use all points, then one has to use the command points2finetriangs. If one is only interested in the number of triangulations, then one can use the commands points2ntriangs and points2nfinetriangs. For d = 2, it s known that the flip graph is connected, so by exploring the flip graph component of a triangulation one can find all triangulations. For d = 3 and d = 4, the problem of deciding wether the flip graphs of all point configurations in R d are connected is still open. For d 5, there are point configurations for which the flip graph is not connected. If the flip graph of a point configuration is not connected, then finding all triangulations gets more involved. We refer the interested reader to [1]. The command to get all triangulations is points2alltriangs. If one is only interested in the number of triangulations or the triangulations using all points, then one can use the commands points2nalltriangs, points2allfinetriangs and points2nallfinetriangs. 6

7 5 Further functionality of TOPCOM As we mentioned before, the chirotope contains all the necessary information. In TOP- COM, it s possible to work directly with the chirotope. In order to do so, one has to save the chirotope of a point configuration in a txt-file and replace points in each command with chiro. TOPCOM includes some commands to generate standard examples, for example cube d generates the d-dimensional cube and cyclic n d generates the cyclic d-polytope with n vertices. To use multiple commands at the same time, one has to connect them by a vertical bar. For example, points2chiro < example.txt chiro2nalltriangs computes the number of all triangulations of our example. The degree of a vertex in the flip graph of the cyclic 2-polytope with 6 vertices can be computed by For the complete list of commands, see [2]. References cyclic 6 2 points2nflips. [1] J. Rambau, TOPCOM: Triangulations of Point Configurations and Oriented Matroids, Mathematical software (Beijing, 2002), pages World Sci. Publ., River Edge, NJ, 2002 [2] [3] J. A. de Loera, S. Hoşten, F. Santos, and B. Sturmfels. The polytope of all triangulations of a point configuration. Documenta Mathematika, 1: , 1996 [4] Francisco Santos, Triangulations of polytopes: personales.unican.es/santosf/talks/icm2006.pdf 7