## On Knots and Cohomology and Dowker

Something about knots and something about topological data analysis and something about the general frame work to do mathematics in a finite setting.

Something about knots and something about topological data analysis and something about the general frame work to do mathematics in a finite setting.

A bit the bigger picture about the mathematical and data structures which come in when working on these finite geometries.

A youtube presentation of May 13, 2023. We point out that we have a relatively simple approach to Eilenberg-Steenrod.

Over the winter break I started to look at Mayer-Vietoris type rules when looking at cohomology of subsets of a simplicial complex. See January 28, 2023 (Youtube) , and February 4th 2023 (Youtube) and most recently on February 19, 2023 (Youtube). Classically, cohomology is considered for simplicial complexes and especially …

A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.

We report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.

The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting things up more generally and there is also some mentioning in the article: allowing general rings and not just division algebras extends the …

The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.

If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.

The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.