Triangle groups, automorphic forms, and torus knots

TL;DR

This paper explores the deep connections between triangle groups, automorphic forms, and torus knots, aiming to explicitly construct diffeomorphisms between knot complements and Lie group quotients using automorphic forms.

Contribution

It provides an explicit construction of diffeomorphisms for torus knot complements via automorphic forms, extending known results for the trefoil knot to a broader class.

Findings

Explicit diffeomorphism construction for torus knot complements

Connection established between automorphic forms and knot topology

Extension of known results from trefoil to general torus knots

Abstract

This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial knots whose complements admit transitive Lie group actions. In fact S^3\K_{p,q} is diffeomorphic to a coset space of the universal covering group of PSL_2(R) with respect to a discrete subgroup G contained in the preimage of a (p,q,\infty)-triangle Fuchsian group. The existence of such a diffeomorphism between is known from a general topological classification of Seifert fibred 3-manifolds. Our goal is to construct an explicit diffeomorphism using automorphic forms. Such a construction is previously known for the trefoil knot K_{2,3} and in fact S^3\K_{2,3} = SL_2(R)/SL_2(Z). The connection between the two sides of the diffeomorphism comes via singularities of plane curves.