Modular cocycles and linking numbers

Abstract

Abstract. It is known that the 3-manifold SL(2, Z)\ SL(2, R) is diffeomorphic to the complement of the trefoil knot in S 3 . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind's eta function under SL(2, Z). In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

Computing intersections of closed geodesics on the modular curve

“…Since the association q → γ q is bijective between PIBQFs and hyperbolic matrices with positive trace in SL(2, Z) (see Section 2.1), all of the linking numbers considered in [DIT17] arise as intersection numbers of modular geodesics.…”

“…We first introduce the links considered in [DIT17]. Let γ ∈ SL(2, Z) be a primitive hyperbolic matrix with positive trace, and recall the matrix M γ as defined in Section 1.1, which satisfied…”

“…His answer is expressed in terms of the Rademacher function, which is directly related to the classical Dedekind η function. In the paper "Linking Numbers and Modular Cocycles" by Duke, Imamoḡlu, and Tóth ( [DIT17]), they consider the linking number of two distinct links [σ] and [γ]. They produce similar results to Ghys, by relating their answer to a modular cocyle (as opposed to a modular form).…”

“…Ideally, the removed trefoil would be filled in, and the linking number of [γ + ] with [σ + ] would be computed. In this paper, we address the first point by relating the linking numbers of [DIT17] to intersection numbers of modular geodesics, and then to combinatorial data coming from Conway's topograph. Using this new interpretation, a conjecture for the unknown linking number is also given.…”

Asymptotic bounds for cycle integrals of the j-function on Markov geodesics

“…Cycle integrals have been related to mock modular forms [11], to modular knots [13] and to class numbers of real quadratic fields [14]. Moreover, cycle integrals of the Klein invariant j share several analogies with singular moduli (the values of the j-function at imaginary quadratic irrationalities) when both are gathered in 'traces' (see [11], [12], [10], [20]).…”