Combinatorial zeta functions counting triangles
Geometric Topology
2025-05-02 v2 Algebraic Topology
Combinatorics
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.
Keywords
Cite
@article{arxiv.2303.11226,
title = {Combinatorial zeta functions counting triangles},
author = {Léo Bénard and Yann Chaubet and Nguyen Viet Dang and Thomas Schick},
journal= {arXiv preprint arXiv:2303.11226},
year = {2025}
}
Comments
26 pages, v2: arguments expanded and in parts simplified, correction of statement of one theorem