A computational framework for weighted simplicial homology

Abstract

We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=F\left[\right[\pi \left]\right]$. This is achieved by starting from a basis for classical homology of the n-th skeleton for the underlying complex with coefficients in the residue field $F$ and then lifting it to a basis for the weighted homology with coefficients in the ring R. Using the latter, a bijection is established between $n+1$ and n dimensional simplices whose weight ratios provide the exponents of the π-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over R. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of R, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.