# Discrete Morse Theory for Weighted Simplicial Complexes

## TL;DR

This paper extends Forman's discrete Morse theory to weighted simplicial complexes, establishing conditions under which weighted homology is preserved during collapses and exploring applications in sequence analysis.

## Contribution

It introduces weighted versions of classical discrete Morse theorems and analyzes how collapses affect weighted homology, with practical applications in sequence analysis.

## Key findings

- Weighted collapses do not always preserve homology.
- Sufficient conditions are identified for homology preservation.
- Application to sequence analysis demonstrates practical relevance.

## Abstract

In this paper, we study Forman's discrete Morse theory in the context of weighted homology. We develop weighted versions of classical theorems in discrete Morse theory. A key difference in the weighted case is that simplicial collapses do not necessarily preserve weighted homology. We work out some sufficient conditions for collapses to preserve weighted homology, as well as study the effect of elementary removals on weighted homology. An application to sequence analysis is included, where we study the weighted ordered complexes of sequences.