Energized simplicial complexes

TL;DR

This paper introduces a novel framework for analyzing simplicial complexes using energy functions and associated matrices, revealing spectral properties, topological invariants, and new algebraic identities.

Contribution

It develops a new energy-based approach to simplicial complexes, connecting spectral, topological, and algebraic properties with novel matrix identities and invariants.

Findings

Determinants of matrices relate to energy functions and products over sets.

Spectral properties of matrices correspond to energy values and topological invariants.

New identities connect energy functions with Euler characteristic and zeta functions.

Abstract

For a simplicial complex with n sets, let W^-(x) be the set of sets in G contained in x and W^+(x) the set of sets in G containing x. An integer-valued function h on G defines for every A subset G an energy E[A]=sum_x in A h(x). The function energizes the geometry similarly as divisors do in the continuum, where the Riemann-Roch quantity chi(G)+deg(D) plays the role of the energy. Define the n times n matrices L=L^--(x,y)=E[W^-(x) cap W^-(y)] and L^++(x,y) = E[W^+(x) cap W^+(y)]. With the notation S(x,y)=1_n omega(x) =delta(x,y) (-1)dim(x) and str(A)=tr(SA) define g=S L^++ S. The results are: det(L)=det(g) = prod_x in G h(x) and E[G] = sum_x,y g(x,y) and E[G]=str(g). The number of positive eigenvalues of g is equal to the number of positive energy values of h. In special cases, more is true: A) If h(x) in -1, 1}, the matrices L=L^--,L^++ are unimodular and L^-1 = g, even if G is a set of sets. B) In the constant energy h(x)=1 case, L and g are isospectral, positive definite matrices in SL(n,Z). For any set of sets G we get so isospectral multi-graphs defined by adjacency matrices L^++ or L^-- which have identical spectral or Ihara zeta function. The positive definiteness holds for positive divisors in general. C) In the topological case h(x)=omega(x), the energy E[G]=str(L) = str(g) = sum_x,y g(x,y)=chi(G) is the Euler characteristic of G and phi(G)=prod_x omega(x), a product identity which holds for arbitrary set of sets. D) For h(x)=t^|x| with some parameter t we have E[H]=1-f_H(t) with f_H(t)=1+f_0 t + cdots + f_d t^d+1 for the f-vector of H and L(x,y) = (1-f_W^-(x) cap W^-(y)(t)) and g(x,y)=omega(x) omega(y) (1-f_W^+(x) cap W^+(y)(t)). Now, the inverse of g is g^-1(x,y) = 1-f_W^-(x) cap W^-(y)(t)/t^dim(x cap y) and E[G] = 1-f_G(t)=sum_x,y g(x,y).