[2211.05957] Number of connected modular knots

[Submitted on 11 Nov 2022]

Download PDF

Overview: The modular group PSL(2;Z) acts on the hyperbolic plane HP with the quotient of the modular surface M, whose unit tangent bundle U is the 3-manifold homeomorphic to the complement of the trefoil knot of the 3-sphere. The hyperbolic conjugate class of PSL(2;Z) corresponds to closed directed geodesics of M. These are lifted into periodic orbits of U geodesic flow that define modular knots. The number of connections between modular knots and trefoils is well understood. In fact, Etienne Ghys showed in his 2006 that the conjugate classes to which they correspond are given by the Rademacher invariants. The Rademacher function is a homogeneous submorphism of his PSL(2;Z), which he recognized in his 1992 with Jean Barge as a half-primitive of the bounded Euler class. This shed light on Michael Atiyah’s 1987 work on the logarithms of his Dedekind eta functions, which he identified with six other important functions that appear in various fields of mathematics. We are interested in the number of connections between modular knots, and derive several formulas with arithmetic, combinatorial, topological, and group-theoretic flavors. In particular, it associates a pair of modular knots with a function defined by the letter diversity of PSL(2;Z). The boundary point limit of this function recovers the bond count. Furthermore, we show that the connectivity of a modular knot minus its reciprocal yields a homogeneous quasimorphism of the modular group, and how to extract the free basis from these. For this, we prove that the link pairing is non-degenerate.