Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

Area of Science:

• Number Theory
• Topology
• Mathematical Physics

Background:

• The Rademacher symbol, a ubiquitous function, relates to modular knots and trefoil knots.
• Previous work by É. Ghys established a connection between linking numbers of modular knots and the Rademacher symbol for specific cases.

Purpose of the Study:

• To generalize the Rademacher symbol for triangle groups $$ for coprime integers (p, q).
• To extend Ghys's theorem to modular knots associated with "missing" torus knots in $$ and lens spaces.

Main Methods:

• Utilizing the theory of harmonic Maass forms for the triangle group $$.
• Introducing and characterizing a generalized Rademacher symbol $$.

Main Results:

• The Rademacher symbol $$ is introduced and characterized for $$.
• Ghys's theorem is generalized for modular knots and "missing" torus knots in $$ and lens spaces.

Conclusions:

• The study successfully generalizes the Rademacher symbol and Ghys's theorem.
• This work bridges concepts in number theory, knot theory, and the theory of modular forms.