作者：[德]迈克·格斯
页数：254
出版社：哈尔滨工业大学出版社
出版日期：2021
ISBN：9787560394237
电子书格式：pdf/epub/txt
内容简介
Imagine two triangles in the three-dimensional space， such that an edge of the one pierces through the interior of the other， and vice versa. In such a geometrical situation， any continuous transformation that separates the two triangles would lead to an intersection of their boundaries at one moment， and so we call the two triangles and their boundaries linked （germ： “verschlungene Dreiecke”）. It is a known fact in graph theory [8] that any embedding of the complete graph with 6 vertices K6 into R3 has at least one pair of those linked triangles. Prof.Dr.U.Brehm （TU-Dresden）， who was my advisor during this diploma thesis， used the so called Gale diagrams to proof that any straight line embedding of the K6 contains either one or exactly three pairs of linked triangles. In Section 1.3.1 we will explain this technique， which leads to the proof of the corresponding Theorem 1.4， and we give visual examples for both cases in Figure 2.
目录
1 Simplices with linked Boundaries and Linking Structures
1.1 Basics of Affine and Convex Geometry
1.2 Simplices with linked Boundaries
1.2.1 (d+4)-Point Configurations
1.3 Simplices with Vertices in a d +3-Point Configuration in Rd
1.3.1 Gale Transform,Gale Diagram, Gale Order
1.3.2 Radon Partitions and Gale Orders
1.4 Properties of d + 3-Point Configuration induced Linking Structures
1.5 Abstract Linking Structures
1.5.1 Cyclic Linking Structure
1.6 Classification of Linking Structures on d +3 Vertices
1.6.1 Introduction .
1.6.2 Integer Partitions
1.6.3 Prime Numbers
1.6.4 Powers of Two
1.6.5 Calculating the Number using Computers
1.6.6 A complete Formula using Pólya Theory of Counting
2 Linking Numbers and oriented Linking Structures
2.1 Oriented Simplices
2.2 Intersection Number
2.3 Linking Number.
2.4 Intersection and Linking Numbers as Concept in Topology
2.5 Oriented Linking Structures induced by Point Configurations
3 Oriented Matroids and abstract oriented Linking Structures
3.1 Introduction to Matroids
3.2 Introduction to oriented Matroids
3.3 Oriented Simplices and their Intersection in oriented Matroids
3.3.1 Linked Boundaries in oriented Matroids and induced Linking Structures
3.4 Dual oriented Matroids
3.5 Oriented Matroids on r+2 vertices and abstract Gale orders
3.6 Realizability of oriented Matroids
3.7 Intersection Numbers and Linking Numbers in oriented Matroids
3.8 Oriented Matroid induced oriented Linking Structures
3.9 Abstract oriented Linking Structures
3.10 Propositions on Linking Structures and oriented Linking Structures.
4 Algorithms and Data Structures
4.1 Linking Matrices
4.1.1 Isomormorphisms of oriented Linking Structures
4.2 Normal Form and Absolute Normal Form of Linking Matrices
4.3 Non-realizable Linking Structures
4.3.1 Checking Linking Structure Axioms
4.3.2 Reorientation Classes of Linking Structures
4.3.3 Exchange Operation
4.3.4 Exchange Operation Graph
5 Computer Generated Data and the Software Package
5.1 Point Configurations
5.1.1 Point Configurations in R3
5.1.2 Point Configurations in R5
5.2 Exchange Operation Graph Results.
