# Combinatorial classification of quantum lens spaces

## TL;DR

This paper provides a combinatorial classification of quantum lens spaces, determining how their isomorphism classes depend on parameters through path counting in associated graphs.

## Contribution

It introduces a combinatorial approach to classify quantum lens spaces based on $SL$-equivalence of integer matrices, linking algebraic classification to graph path counting.

## Key findings

- Determines the dimension bounds for quantum lens spaces based on primary parameters.
- Establishes a combinatorial method for classification using graph path counting.
- Connects $SL$-equivalence of matrices to isomorphism classes of quantum lens spaces.

## Abstract

We answer the question of how large the dimension of a quantum lens space must be, compared to the primary parameter $r$, for the isomorphism class to depend on the secondary parameters. Since classification results in C*-algebra theory reduces this question to one concerning a certain kind of $SL$-equivalence of integer matrices of a special form, our approach is entirely combinatorial and based on the counting of certain paths in the graphs shown by Hong and Szyma\'nski to describe the quantum lens spaces.