# Rational Dyck Paths in the Non Relatively Prime Case

## TL;DR

This paper explores the combinatorial structure of rational Dyck paths when the slope parameters are not coprime, extending previous work and establishing new bijections relevant to algebraic geometry and knot theory.

## Contribution

It generalizes the relationship between rational Dyck paths and invariant subsets to the non relatively prime case, introducing new bijections and combinatorial frameworks.

## Key findings

- Extended the combinatorial correspondence to non coprime cases
- Established a bijection between $(dn,dm)$-Dyck paths and $d$-tuples of $(n,m)$-Dyck paths
- Laid groundwork for linking rational Catalan combinatorics with affine Springer fibers and knot invariants

## Abstract

We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb Z,$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$--Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.