Laplacian polytopes of simplicial complexes

TL;DR

This paper introduces Laplacian polytopes for simplicial complexes, studies their properties, and characterizes their structure for boundary complexes of simplices, revealing connections to cyclic polytopes and unimodular triangulations.

Contribution

It extends Laplacian simplices to higher-dimensional complexes, characterizes the polytopes for boundary complexes of simplices, and provides explicit triangulations and volume properties.

Findings

For odd d, the d-th Laplacian polytope is a (d+1)-simplex.

For even d, it is combinatorially equivalent to a d-dimensional cyclic polytope.

Explicit unimodular triangulations enable volume and polynomial root analysis.

Abstract

Given a (finite) simplicial complex, we define its $i$-th Laplacian polytope as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the $d$-th Laplacian polytope of the boundary of a $(d+1)$-simplex $\partial(\sigma_{d+1})$. If $d$ is odd, then as for graphs, the $d$-th Laplacian polytope turns out to be a $(d+1)$-simplex in this case. If $d$ is even, we show that the $d$-th Laplacian polytope of $\partial(\sigma_{d+1})$ is combinatorially equivalent to a $d$-dimensional cyclic polytope on $d+2$ vertices. Moreover, we provide an explicit regular unimodular triangulation for the $d$-th Laplacian polytope of $\partial(\sigma_{d+1})$. This enables us to to compute the normalized volume and to show that the $h^\ast$-polynomial is real-rooted and unimodal, if $d$ is odd and even, respectively.