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Arithmetic knots and links

Posted in
Alan Reid
Rice University/MPIM
Mon, 02/05/2022 - 15:00 - 16:00
Parent event: 
MPIM Topology Seminar

Hybrid talk.
Contact for zoom details: Barthel, Ozornova, Ray, Teichner.

A two-part mini-series in the MPIM topology seminar
Series title: Arithmeticity of link complements

Going back to the work of Thurston, "arithmetic" link complements have played an important role in the development of geometric structures on 3-manifolds. From a different perspective, such manifolds have attracted the attention of number theorists through their lack of cuspidal cohomology.  Indeed, a key early result in understanding arithmetic link complements has a strong Bonn connection through work of Harder, Grunewald, Schwermer.. that eventually led to the resolution of the Cuspidal Cohomology Problem by K. Vogtmann, the upshot being there are only finitely many commensurability classes of arithmetic link complements.

This series of lectures will:

(a) survey some of this;
(b) talk about work of Baker-Goerner-Reid, and and open problem concerning congruence arithmetic link complements;
(c) construct infinitely many alternating arithmetic link complements all commensurable with the figure-eight knot complement (thereby answering questions of D. Futer and independently M. Lackenby).

Talk 1, Alan Reid: Arithmetic knots and links.

Abstract: Let $M$ be a closed orientable 3-manifold. A link $L \subset M$ is called arithmetic if $M\setminus L$ is commensurable with a Bianchi orbifold $Q_d=H^3/PSL(2,O_d)$. In this introductory talk we discuss various questions about the geometry, topology and number theory of these manifolds.



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