Discrete Morse theory for weighted simplicial complexes

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.2010 Mathematics Subject Classification. Primary 55N35; Secondary 55U10.

Political structures and the topology of simplicial complexes

“…It is not clear how to extend the results of this section to weighted complexes. There is a version of homology for this situation [11,46] that could generalize our observations to weighted complexes, but it requires the divisibility of the weights with respect to face inclusions.…”

Political structures and the topology of simplicial complexes

“…It is not clear how to extend the results of this section to weighted complexes. There is a version of homology for this situation [11,46] that could generalize our observations to weighted complexes, but it requires the divisibility of the weights with respect to face inclusions.…”

Persistent Topological Laplacians—A Survey

“…Many theories and models involving weighted simplicial complexes exist (e.g., [ 83 – 88 ]). Here, we focus on the theory of weighted simplicial complexes proposed by Dawson [ 89 ] and later developed in [ 90 – 96 ]. A weighted simplicial complex is a simplicial complex where each simplex has a weight valued in a commutative ring , such that if , then is divisible by .…”

Persistent Topological Laplacians -- a Survey

“…Here we focus on the theory of weighted simplicial complexes proposed by Robert J. MacG. Dawson [44] and later developed in [18,19,94,124,126,155,156]. A weighted simplicial complex is a simplicial complex where each simplex σ has a weight w(σ) valued in a commutative ring R, such that if σ τ , then w(τ ) is divisible by w(σ).…”

Political structures and the topology of simplicial complexes

“…It is not clear how to extend the results of this section to weighted complexes. There is a version of homology for this situation [11,46] that could generalize our observations to weighted complexes, but it requires the divisibility of the weights with respect to face inclusions.…”

Persistent Topological Laplacians—A Survey

“…Many theories and models involving weighted simplicial complexes exist (e.g., [ 83 – 88 ]). Here, we focus on the theory of weighted simplicial complexes proposed by Dawson [ 89 ] and later developed in [ 90 – 96 ]. A weighted simplicial complex is a simplicial complex where each simplex has a weight valued in a commutative ring , such that if , then is divisible by .…”

Persistent Topological Laplacians -- a Survey

“…Here we focus on the theory of weighted simplicial complexes proposed by Robert J. MacG. Dawson [44] and later developed in [18,19,94,124,126,155,156]. A weighted simplicial complex is a simplicial complex where each simplex σ has a weight w(σ) valued in a commutative ring R, such that if σ τ , then w(τ ) is divisible by w(σ).…”

Political structures and the topology of simplicial complexes

“…It is not clear how to extend the results of this section to weighted complexes. There is a version of homology for this situation [11,46] that could generalize our observations to weighted complexes, but it requires the divisibility of the weights with respect to face inclusions.…”

Persistent Topological Laplacians—A Survey

“…Many theories and models involving weighted simplicial complexes exist (e.g., [ 83 – 88 ]). Here, we focus on the theory of weighted simplicial complexes proposed by Dawson [ 89 ] and later developed in [ 90 – 96 ]. A weighted simplicial complex is a simplicial complex where each simplex has a weight valued in a commutative ring , such that if , then is divisible by .…”

Persistent Topological Laplacians -- a Survey

“…Here we focus on the theory of weighted simplicial complexes proposed by Robert J. MacG. Dawson [44] and later developed in [18,19,94,124,126,155,156]. A weighted simplicial complex is a simplicial complex where each simplex σ has a weight w(σ) valued in a commutative ring R, such that if σ τ , then w(τ ) is divisible by w(σ).…”