Algorithmica

, Volume 21, Issue 4, pp 331–346

Computing Betti Numbers via Combinatorial Laplacians

  • J. Friedman

DOI: 10.1007/PL00009218

Cite this article as:
Friedman, J. Algorithmica (1998) 21: 331. doi:10.1007/PL00009218
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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, ν , of eigenvalues which we have yet to understand fully.

We numerically verify a conjecture of Bj{ö}rner, Lov{á}sz, Vre{\'c}ica, and {\u Z}ivaljevi{\'c} 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 rigorous (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.

Key words. Betti numbers, Combinatorial Laplacian, Power method, Simplicial complexes, Chessboard complexes

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • J. Friedman
    • 1
  1. 1.Department of Mathematics, 1984 Mathematics Road, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2. jf@math.ubc.ca.CA