Computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology. (English) Zbl 1498.55006

Suppose that \(M\) is a bigraded persistence module over the field \(K\). That is, \(M\) is a direct sum of \(K\)-vector spaces \(M_{\mathbf{z}}\) indexed by pairs of integers \(\mathbf{z}=(z_1,z_2)\), and is equipped with a \(K[x_1,x_2]\)-module structure such that \(x_1^ix_2^jM_{(z_1,z_2)}\) lies in \(M_{(z_1+i,z_2+j)}\). Let \(I\!M\) denote the submodule generated by elements of the form \(pm\), where \(p\) is a monomial of positive degree and \(m\) is in \(M\). Given a left resolution \(\cdots\xrightarrow{\partial_3}F_2\xrightarrow{\partial_2}F_1\xrightarrow{\partial_1}F_0 \rightarrow M\rightarrow 0 \) by free bigraded persistence modules such that the image of \(\partial_i\) is contained in \(I\!F_{i-1}\), the \(i\)-th bigraded Betti number \(\beta_i(\mathbf{z})\) of \(M\) at grade \(\mathbf{z}\) is taken to be the \(K\)-vector space dimension of \((F_i/I\!F_i)_{\mathbf{z}}\). In the case when there is a chain complex \(X\xrightarrow{f}Y\xrightarrow{g}Z\) of free bigraded persistence modules with \(M=\ker\,g/\mathrm{im}\,f\), the authors of the article provide an algorithm for computing the \(0\)-th and \(1\)-st bigraded Betti numbers of \(M\). Correctness of the algorithm is shown, and spacial and computational efficiency analyses are given. Numerical experiments performed indicate that the algorithm significantly outperforms standard methods.


55N31 Persistent homology and applications, topological data analysis
13D02 Syzygies, resolutions, complexes and commutative rings
Full Text: DOI arXiv


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