# In topological data analysis, do bar codes that begin and end at the same index mean anything?

The typical workflow in topological data analysis is from point cloud data to filtration to a list of bar codes corresponding to each dimension.

A filtration is a sequence of simplicial complexes, each one a subset of the next; where a simplicial complex is a gluing of many simplices such that the intersection of any two simplices (if it exists) is a simplex. Each simplicial complex can be obtained from the point cloud data by, for example, taking a sequence of distances and connecting points which fall within that distance to form a graph, then identifying complete subgraphs as simplices to make a simplicial complex. This is called the Vietoris-Rips complex. Another common way to construct a filtration is the Cech complex, which makes a simplex out of a set of points iff balls of a certain radius intersect.

The filtration is the input to the algorithm which computes persistent homology. Simplicial homology is the simplest of homology theories and it allows one to identify connected components, tunnels, hollow volumes, and higher dimensional cavities in a simplicial complex. Persistent homology does the same for filtrations, except it also measures how long each cluster/hole lasts in the sequence of complexes. The computation is a bit tricky, to learn about it I would refer you to the paper that showed that the computation is actually practical. The output of persistent homology is a bar code diagram for each dimension. A bar code has a beginning, which is the index in the filtration where the cluster/hole begins, and it might have an end where the cluster/hole dies.

So it seems natural to disregard bar codes that have the same beginning and end; they don't persist at all. Even if a feature is present only in the ith complex in the filtration, it will have the bar code (i, i+1) because it dies in the (i+1)th complex.

However, I want to be completely sure because I'm maintaining a TDA package. I just finished a patch and was considering adding an extra feature where these apparently meaningless bar codes are automatically filtered out.

• From here it seems that the barcodes are the outcome of the analysis. Perhaps you can expand your question by a few more paragraphs, explaining more of the situation? – Yuval Filmus Jul 30 '18 at 8:31
• YuvalFilmus do my editions suffice? – Eben Kadile Jul 30 '18 at 8:47
• Thanks, that's much better. Hopefully someone knowledgeable would reply. – Yuval Filmus Jul 30 '18 at 8:49