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The example at the beginning of the video https://youtu.be/qGkIuJmXhts,

(Filtration of the example),

I have a question about the barcode of a 1-dim persistent barcode. The video's author gave two homology bars, one born 2 died 5, and the other born 3 died 4. So why not consider the loop, the lower-left triangle, "a d c " born 3 died 5.

[Q1]Although I know that the barcode given by the author of the video is correct because the number of holes corresponding to each index of filtration is correct, I still can't understand why the triangle in the lower-left corner is not considered.

In addition, fig.5 in https://appliednetsci.springeropen.com/articles/10.1007/s41109-019-0179-3 has the same situation, fig.6 shows the 1-dim persistent barcode,

the filtration of the other example,barcode

For this example, I have a different opinion, I think the 1-dim persistent barcode should be:

my opinion

[Q2]The number of holes corresponding to each index of filtration is correct, so whether there are different barcodes for the filtration of a complex.

Thanks for any help.

Billal
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1 Answers1

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Regarding the filtration in that YouTube video, the loop given by the edges cd+da+ac does not give another bar because it is already represented as a linear combination of the other two generators that are already present (remember homology is linear). Indeed, a generator for the 1-dimensional bar that begins at 2 is ab+bc+cd+da. A generator for the 1-dimensional bar that begins at 3 is ab+bc+ca. Your proposed extra generator is redundant, i.e. is a linear combination of the other generators already described. Indeed, with Z/2Z coefficients we have that (ab+bc+cd+da)+(ab+bc+ca)=cd+da+ca=cd+da+ac. So your proposed new generator is redundant; it is a sum of the two generators already given.

For Figure 5 of https://appliednetsci.springeropen.com/articles/10.1007/s41109-019-0179-3 I agree the barcodes given look incorrect and the barcodes you give look correct, at least to me at first sight.

Henry Adams
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