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Infinitely many arithmetic alternating links

Posted in
Mark Baker
Université de Rennes 1
Mon, 02/05/2022 - 16:30 - 17:30
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 2, Mark Baker: Infinitely many arithmetic alternating links.

Abstract: In this talk we answer affirmatively a question asked independently by D. Futer and M. Lackenby as to whether there are infinitely many arithmetic alternating link complements in $S^3$. The construction is explicit, using ideas that have their origins in Thurston's Notes and work of Hatcher.

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