Computing Betti Numbers via Combinatorial Laplacians
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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.