Studying knots in self-covers of the modular flow

TL;DR

This paper introduces a combinatorial tool to analyze topological properties of modular knots by constructing templates for infinite Anosov flows on the trefoil complement, enabling detailed study of knot properties and link families.

Contribution

It provides a method to lift modular templates via self-coverings, creating a framework to analyze knots and links in these flows with explicit constructions.

Findings

Constructed templates for infinitely many Anosov flows on the trefoil complement.

Enabled analysis of knot properties of closed geodesics in these flows.

Explicitly constructed an infinite family of links with specific properties.

Abstract

In this paper we provide a combinatorial tool to help study some topological properties of modular knots. We construct templates for the infinitely many Anosov flows on the trefoil complement, which are lifts of the geodesic flow on the modular surface, by lifting Ghys' modular template using self-covering of the trefoil complement of order $6k+1$, for $k\in \mathbb{N}_{>0}$. This allows to study the knot properties of closed geodesics in these flows, and an explicit construction of an infinite family of links of two components with one of them being the trefoil, all commensurable to one another.