Arithmetic and Topology of Modular Knots

Meeting Details

For more information about this meeting, contact Donna Cepullio, Sergei Tabachnikov.

Speaker: Christopher-Lloyd Simon, Penn State

Abstract: We study several arithmetic and topological structures on the set of conjugacy classes of the modular group PSL(2;Z), such as equivalence relations or bilinear functions. The modular group PSL(2;Z) acts on the hyperbolic plane with quotient the modular orbifold M, whose oriented closed geodesics correspond to the hyperbolic conjugacy classes in PSL(2;Z). For a field K containing Q, two matrices of PSL(2;Z) are said to be K-equivalent if they are conjugated by an element of PSL(2;K). For K=C this amounts to grouping modular geodesics of the same length. For K=Q we obtain a refinement of this equivalence relation, and we will give a geometrical interpretation in terms of the modular geodesics (angles at the intersection points and lengths of the ortho-geodesics). The unit tangent bundle U of the modular orbifold M is a 3-dimensional manifold homeomorphic to the complement of trefoil in the sphere. The modular knots in U are the periodic orbits for the geodesic flow, lifts of the closed oriented geodesics in M, and also correspond to the hyperbolic conjugacy classes in PSL(2;Z). Their linking number with the trefoil is well understood and we are interested in the linking numbers between two modular knots. To a pair of modular knots, we associate a function defined on the character variety of of PSL(2;Z), whose limit at the boundary recovers their linking number.

---

Room Reservation Information

Room Number: 114 McAllister

Date: 10/20/2022

Time: 3:30pm - 4:30pm