The Burau representation and shapes of polyhedra

Dlugie, Ethan

doi:10.2140/agt.2024.24.2787

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The Burau representation and shapes of polyhedra

Ethan Dlugie

Algebraic & Geometric Topology 24 (2024) 2787–2805

DOI: 10.2140/agt.2024.24.2787

Abstract

We use a geometric approach to show that the reduced Burau representation specialized at roots of unity has another incarnation as the monodromy representation of a moduli space of Euclidean cone metrics on the sphere, as described by Thurston. Using the theory of orbifolds, we leverage this connection to identify the kernels of these specializations in some cases, partially addressing a conjecture of Squier. The $4$ –strand case is the last case where the faithfulness question for the Burau representation is unknown, a question that is related eg to the question of whether the Jones polynomial detects the unknot. Our results allow us to place the kernel of this representation in the intersection of several topologically natural subgroups of $B_{4}$ .

Keywords

braid groups, Burau representation, roots of unity, flat surfaces, shapes of polyhedra

Mathematical Subject Classification

Primary: 20F36, 57K20

Secondary: 57R18

References

Publication

Received: 12 October 2022

Revised: 31 May 2023

Accepted: 3 July 2023

Published: 19 August 2024

Authors

Ethan Dlugie
Department of Mathematics University of California, Berkeley Berkeley, CA United States

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Volume 24, issue 5 (2024)