Simplicial Complex Analysis: Tetrahedron Boundary Exploration
Explore various aspects of a simplicial complex representing the boundary of a tetrahedron, including finding vertices, matrices, ranks, and barcodes for homology groups. Understand and interpret the results for different dimensions of the complex.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: v2 C0 = C1 = v4 C2 = v1 v3 C3 = Z0 = Explain your answer for Z0: B0 = H0 = Rank C0 = Rank Z0 = Rank B0 = Rank H0 = |C0| = |Z0| = |B0| = |H0|=
Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: v2 Find the matrix for 0: v4 Find the matrix for 1: v1 v3 Simplify the matrix for 1 so that the nonzero columns are linearly independent. Write the basis element above its corresponding column.
Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: v2 Find the matrix for 2: v4 v1 v3 Simplify the matrix for 2 so that the nonzero columns are linearly independent. Write the basis element above its corresponding column.
Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: v2 Z1 = v4 Explain your answer for Z1: v1 v3 B1 = H1 = Rank C1 = Rank Z1 = Rank B1 = Rank H1 = |C1| = |Z1| = |B1| = |H1|=
Let C be the simplicial complex on the right (the boundary of a tetrahedron). Find the following: v2 Z2 = v4 Explain your answer for Z2: v1 v3 B2 = H2 = Rank C2 = Rank Z2 = Rank B2 = Rank H2 = |C2| = |Z2| = |B2| = |H2|=
Barcode for H0 Barcode for H1
