Combinatorial zeta functions counting triangles

TL;DR

This paper introduces combinatorial zeta functions that count geodesic paths in triangulated manifolds, revealing topological invariants like Betti numbers and linking numbers through random walk analysis.

Contribution

It establishes a novel connection between combinatorial zeta functions and topological invariants of manifolds using higher-dimensional skeleta and random walks.

Findings

Recover the first Betti number of manifolds.

Determine L2-Betti numbers via zeta functions.

Compute linking numbers of knots in 3-manifolds.

Abstract

In this paper, we compute special values of certain combinatorial zeta functions counting geodesic paths in the (n-1)-skeleton of a triangulation of a n-dimensional manifold. We show that they carry a topological meaning. As such, we recover the first Betti number and L2-Betti number of compact manifolds, and the linking number of pairs of null-homologous knots in a 3-manifold. The tool to relate the two sides (counting geodesics/topological invariants) are random walks on higher dimensional skeleta of the triangulation.