Before revision:  The method relies on implicit representations of the coboundary operator and of the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix  .
After revision:  The method relies on implicit representations of the coboundary operator and of the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix . Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest .