[Paper Review] The counting matrix of a simplicial complex

Oliver Knill|arXiv (Cornell University)|Jul 22, 2019

Topological and Geometric Data Analysis13 references4 citations

TL;DR

This paper introduces the counting matrix K of a finite abstract simplicial complex G, where K(x,y) counts the number of common subsimplices in sets x and y. It proves K is unimodular with integer entries (in SL(n,Z)), positive definite, and its inverse has a Green's function formula involving stars and signs; the spectrum of K and K⁻¹ are identical, leading to spectral symmetry and a functional equation for the zeta function.

ABSTRACT

For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1) always agree so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets in G. In comparison, we had in the connection matrix case the identity that the sum of the matrix elements of L^(-1) is the Euler characteristic of G.

Motivation & Objective

• To define and analyze a new matrix, the counting matrix K, associated with a finite abstract simplicial complex G.
• To establish that K is always unimodular and belongs to SL(n,Z), extending properties known for the connection matrix.
• To derive a closed-form expression for the inverse of K using star sets and Eulerian signs.
• To prove that the spectra of K and K⁻¹ are identical, implying spectral symmetry.
• To show that the energy theorem for K relates the sum of entries of K⁻¹ to the total number of simplices in G.

Proposed method

• Define the counting matrix K(x,y) as the number of subsimplices common to x and y in a finite simplicial complex G.
• Prove that K is always positive definite using combinatorial and algebraic arguments.
• Establish that K ∈ SL(n,Z) by showing det(K) = 1 and integer entries via inclusion-exclusion and simplicial structure.
• Derive the inverse matrix formula: K⁻¹(x,y) = w(x)w(y)|W⁺(x) ∩ W⁺(y)|, where w(x) = (−1)^dim(x) and W⁺(x) is the star of x.
• Use spectral symmetry to show spec(K) = spec(K⁻¹), implying that Q = K − K⁻¹ satisfies spec(Q) = −spec(Q).
• Define the zeta function z(s) = ∑ λ⁻ˢ over eigenvalues λ of K, and prove the functional equation z(a+ib) = z(−a+ib).

Experimental results

Research questions

• RQ1 What algebraic and spectral properties does the counting matrix K of a simplicial complex possess?
• RQ2 How does the inverse of K relate to the combinatorial structure of the complex, particularly through stars and signs?
• RQ3 Does the spectrum of K exhibit symmetry, and if so, what functional equation does the associated zeta function satisfy?
• RQ4 How does the energy theorem for K compare to the classical energy theorem for the connection matrix L?
• RQ5 Can the counting matrix be shown to be unimodular and positive definite for all finite simplicial complexes?

Key findings

• The counting matrix K is always unimodular and belongs to SL(n,Z), ensuring integer entries and determinant 1.
• The inverse of K is given explicitly by K⁻¹(x,y) = w(x)w(y)|W⁺(x) ∩ W⁺(y)|, where w(x) = (−1)^dim(x) and W⁺(x) is the star of x.
• The matrix K is positive definite for all finite abstract simplicial complexes.
• The spectra of K and K⁻¹ are identical, leading to spectral symmetry: spec(Q) = −spec(Q) for Q = K − K⁻¹.
• The zeta function z(s) = ∑ λ⁻ˢ over eigenvalues λ of K satisfies the functional equation z(a+ib) = z(−a+ib).
• The energy theorem states that the sum of all entries of K⁻¹ equals the total number of simplices in G.