Linking number of modular knots

TL;DR

This paper calculates the linking number of modular knots in a specific space, answering a question from 2007, by relating modular links to Lorenz links and using intersection theory.

Contribution

It provides the first explicit computation of linking numbers of modular knots, connecting modular and Lorenz links through novel intersection number techniques.

Findings

Computed linking numbers of modular knots in the specified space.

Established a correspondence between modular and Lorenz links.

Compared two different formulas for linking numbers, validating their consistency.

Abstract

We compute the linking number of two modular knots in the space $\text{PSL}(2, \mathbb{Z})\backslash\text{PSL}(2, \mathbb{R})$ with the trefoil filled in, which answers a question posed by Ghys in 2007. This computation is realized through the correspondence between modular links and Lorenz links, and can be thought of as an intersection number involving Conway topographs. We compare this to a second formula for the linking number of Lorenz links, which was proven by Stephen F. Kennedy in 1994.