Department of Mathematics

Geometry and Topology

  •  Amitesh Datta, Princeton University
  •  Does the Jones polynomial of a knot detect the unknot? A novel approach via braid group representations and class numbers of number fields.
  •  02/07/2023
  •  3:30 PM - 4:30 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Peter Kilgore Johnson (

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation. In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s and is equivalent to whether B_4 is a group of 3 x 3 matrices). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones. I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.



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