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Combinatorial zeta functions counting triangles

Léo Bénard, Yann Chaubet, Nguyen Viet Dang, Thomas Schick

TL;DR

The paper introduces combinatorial zeta functions that count primitive closed geodesics in the (n-1)-skeleton of a triangulated n-manifold and establishes a deep link between geodesic counting and topology via a signed geodesic random walk encoded by a transfer matrix. In the compact case, the zeta function satisfies $\zeta_{\mathscr T}(z) = \det(\mathrm{Id} - z T)$ and vanishes to order $b_1(M)$ at $z = 1/(n+2)$, enabling recovery of the first Betti number from length data; in the non-compact (L2) setting, the near singular behavior of the $L^2$ zeta function recovers the first $L^2$-Betti number $b_1^{(2)}(M,\pi)$, with determinant-class conditions linking to $\det_{FK}$. The authors also define a combinatorial Poincaré series for orthogeodesic paths between null-homologous knots, showing it evaluates to the classical linking number at $z = 1/(n+2)$. Overall, the work connects counting geodesics, spectral properties of the combinatorial Laplacian, and fundamental topological invariants through a transfer-matrix framework, with potential implications for $L^2$-torsion and Novikov-Shubin invariants.

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.

Combinatorial zeta functions counting triangles

TL;DR

The paper introduces combinatorial zeta functions that count primitive closed geodesics in the (n-1)-skeleton of a triangulated n-manifold and establishes a deep link between geodesic counting and topology via a signed geodesic random walk encoded by a transfer matrix. In the compact case, the zeta function satisfies and vanishes to order at , enabling recovery of the first Betti number from length data; in the non-compact (L2) setting, the near singular behavior of the zeta function recovers the first -Betti number , with determinant-class conditions linking to . The authors also define a combinatorial Poincaré series for orthogeodesic paths between null-homologous knots, showing it evaluates to the classical linking number at . Overall, the work connects counting geodesics, spectral properties of the combinatorial Laplacian, and fundamental topological invariants through a transfer-matrix framework, with potential implications for -torsion and Novikov-Shubin invariants.

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.
Paper Structure (10 sections, 15 theorems, 86 equations, 4 figures)

This paper contains 10 sections, 15 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial Laplacian
  4. $L^2$-invariants
  5. First Betti number and combinatorial Ruelle zeta function
  6. The compact case
  7. The non-compact case: $L^2$-Betti numbers
  8. Combinatorial linking number
  9. Examples and final remarks
  10. Concluding remarks

Key Result

Theorem 2

Assume that $M$ is a compact oriented manifold of dimension $n\geqslant 2$ with triangulation $\mathscr T$. The combinatorial zeta function converges for $|z|$ small enough. It is a polynomial function of degree $|\mathscr T^{(n-1)}|$ in $z$ and vanishes of order $b_1(M)$ at $z=(n+2)^{-1}$. Here, $|\mathscr T^{(n-1)}|$ denotes the cardinality of the $(n-1)-$skeleton of the triangulation.

Figures (4)

  • Figure 1: An orthogeodesic $c = (\tau_1, \tau_2, \tau_3, \tau_4)$ linking $\kappa_1$ and $\kappa_2$. The knot $\kappa_1$ is a one-chain in $\mathscr T$ while $\kappa_2$ is a one-chain in $\mathscr T^\vee$.
  • Figure 2: A $2$-dimensional triangulation $\mathscr T$ (in black), together with a dual polyhedral decomposition $\mathscr T^\vee$ (in red).
  • Figure 3: Some closed primitive unmarked geodesics $\gamma_i, \, i=1 \ldots 4,$ from left to right. The extra edges are browsed twice (back-and-forth). For the first one on the left, $n_{\gamma_1} = 0$, then $n_{\gamma_2}=1$ and $n_{\gamma_3} = n_{\gamma_4} = 2$.
  • Figure 4: A permitted $\Delta$-complex decomposition of the torus which is not quite a triangulation. The top and bottom segment have to be identified as well as the left and right one.

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Corollary 4
  • proof
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Definition 9
  • ...and 30 more