Modular knots obey the Chebotarev law

TL;DR

This paper explores the analogy between knots and prime numbers, demonstrating that modular knots follow the Chebotarev law, and refines the construction of planetary links in hyperbolic fibered links with implications for arithmetic topology.

Contribution

It establishes that modular knots around torus knots in S^3 obey the Chebotarev law, extending McMullen's theorem to cusped orbifolds and refining the planetary link construction.

Findings

Modular knots obey the Chebotarev law.

Refinement of planetary link construction in hyperbolic fibered links.

Extension of McMullen's theorem to cusped orbifolds.

Abstract

We study knots which behave like prime numbers. We discuss the planetary link raised from a hyperbolic fibered link in $S^3$ with an emphasis on surgeries, point out certain subtleness, and refine the construction. In addition, we point out a version of McMullen's theorem for the cases over cusped orbifolds and deduce that the family of modular knots around any torus knot $K_{a,b}$ in $S^3$ together with the missing knot $K_{a,b}$ obey the Chebotarev law. Furthermore, we attach several remarks in the view of arithmetic topology.