Computing Betti Numbers via Combinatorial Laplacians
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  • Abstract:

    We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, $\nu$, of eigenvalues which we have yet to fully understand. We numerically verify a conjecture of \bjetal on the chessboard complexes $C(4,6)$, $C(5,7)$, and $C(5,8)$. Our verification suffers a technical weakness, which can be overcome in various ways; we do so for $C(4,6)$ and $C(5,8)$, giving a completely rigourous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.