TOPCOM: Triangulations of Point Configurations and Oriented Matroids - PowerPoint PPT Presentation

Introduction TOPCOM References TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau Clemens Pohle Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Basics A ⊂ R d : d -dimensional configuration of n points k -simplex: sub-configuration of A consisting of k + 1 affinely independent points triangulation: collection T of d -simplices whose convex hulls cover conv ( A ) and intersect properly: ∀ σ, τ ∈ T : conv ( σ ∩ τ ) = conv ( σ ) ∩ conv ( τ ) Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Chirotope � → { + , − , 0 } (gives the orientation � A chirotope: function d +1 of every d + 1-subset of A ) 012 + 013 0 023 - 123 - Fact: chirotope contains all the information we need! Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References What can we do with TOPCOM? Example: Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Compute the chirotope 012 - 013 - 014 + 023 - 024 + 034 + 123 - 124 + 134 + 234 + χ ( i 1 , i 2 , ..., i d +1 ) = sign ( det ( a i 1 , a i 2 , ..., a i d +1 )) Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Construct a Placing Triangulation 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 Start with a d -simplex In each step, add a point a k +1 and all simplices F ∪ a k +1 for which F ∈ F k is visible from a k +1 Stop when F k is empty Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Construct a Placing Triangulation Remarks: The resulting triangulation depends on the numbering of the points It’s possible that not all points are used for the triangulation To get a triangulation using all points ( fine triangulation ), the missing points are added one by one by ‘flipping-in’ Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Flips Flip: exchange between the two possible triangulations in a subset of d + 2 points Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Explore a flip-graph component Flip graph: triangulations as vertices, two triangulations are connected if they differ by a flip Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Explore a flip-graph component The flip-graph of a regular hexagon Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References Check a potential triangulation Theorem Let A be a full-dimensional point configuration in R d . A set T of d-simplices of A is a triangulation of A if and only if For every pair S, S ′ of simplices in T, there exists no circuit ( Z + , Z − ) in A with Z + ⊆ S and Z − ⊆ S ′ (Intersection Property). For each facet of a simplex S in T there is another simplex S ′ � = S having F as a facet, or F is contained in a facet of A (Union Property). Circuit: partition of a minimal affinely dependent set into Z + and Z − such that conv ( Z + ) ∩ conv ( Z − ) � = ∅ Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau

Introduction TOPCOM References References J. Rambau, TOPCOM: Triangulations of Point Configurations and Oriented Matroids, Mathematical software (Beijing, 2002), pages 330-340. World Sci. Publ., River Edge, NJ, 2002 http://www.rambau.wm.uni-bayreuth.de/TOPCOM J. A. de Loera, S. Hos ¸ten, F. Santos, and B. Sturmfels. The polytope of all triangulations of a point configuration. Documenta Mathematika, 1:103–119, 1996 Francisco Santos, Triangulations of polytopes: personales.unican.es/santosf/Talks/icm2006.pdf Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨ org Rambau