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

Introduction TOPCOM References

TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg Rambau

Clemens Pohle

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg Rambau

Introduction TOPCOM References

Basics

A ⊂ Rd: 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¨

• rg Rambau

Introduction TOPCOM References

Chirotope

chirotope: function

A

d+1

→ {+, −, 0} (gives the orientation

• f every d + 1-subset of A)

012 + 013 023

• 123
• Fact: chirotope contains all the information we need!

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg Rambau

Introduction TOPCOM References

What can we do with TOPCOM?

Example:

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg Rambau

Introduction TOPCOM References

Compute the chirotope

012

• 013
• 014

+ 023

• 024

+ 034 + 123

• 124

+ 134 + 234 + χ(i1, i2, ..., id+1) = sign(det(ai1, ai2, ..., aid+1))

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg Rambau

Introduction TOPCOM References

Construct a Placing Triangulation

Ak: set of points that is already triangulated Tk: placing triangulation of Ak Fk: set of all boundary facets of Tk that are interior in A Start with a d-simplex In each step, add a point ak+1 and all simplices F ∪ ak+1 for which F ∈ Fk is visible from ak+1 Stop when Fk is empty

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg 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¨

• rg Rambau

Introduction TOPCOM References

Flips

Flip: exchange between the two possible triangulations in a subset

• f d + 2 points

Clemens Pohle TOPCOM: Triangulations of Point Configurations and Oriented Matroids - J¨

• rg 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¨

• rg 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¨

• rg Rambau

Introduction TOPCOM References

Check a potential triangulation

Theorem Let A be a full-dimensional point configuration in Rd. 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¨

• rg 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¨

• rg Rambau