Date/Time: 3 November 2023, 4:30pm ET
 Location: Elizabethtown College, Esbenshade 382
     Zoom: TBA
  Speaker: Neal Stoltzfus, Louisiana State University
    Title: Cracking the Algebraic Cold Heart of the Braid Group

 Abstract: 
 
The ubiquitous braid group can be approached from many perspectives
(algebraic geometrically, combinatorially, geometric group
theoretically).  This talk will focus on the "arithmetical topological,"
developing ties with algebraic number theory.  We will concentrate on
developing a description of the image of the known injective (finite
dimensional faithful) representation of Lawrence/Krammer/Bigelow.

Recalling Artin's faithful infinite dimensional representation and his
"combing of pure braids", we first develop an analog for the
(unfaithful) Burau representation case using covering spaces, local
coefficients and the Reidemeister homotopical intersection theory for
the braid action on one-point configurations.

Next we introduce the braid action on the (unordered) two-point
configuration space utilized by Krammer & Bigelow. For an easier
description and computation, we will utilize the two-fold covering space
of ordered pair configurations which form a tower of hyperplane
arrangements.  The complements of these (discriminantal) arrangements
are fibrations whose fundamental groups are semi-direct products from
pure braid combing.

Computing Blanchfield duality of the complements of open tubular
neighborhood of the hyperplane arrangements we determine the first
restriction on the image of the LKB representation: Hermitian form
invariance under the intersection form discovered by Budney & Song.
Additional conditions are determined by arithmetic monoidal conditions
arising from matrix invariants over the counting number (N) polynomial
monoids, N[q, 1-q,t] related to (Dehornoy-Paris-)Garside group
structures within the braid group.

We will close with potential applications to classical link
classification through a good algebraic understanding of
stabilization. In addition, we hope to extend the structures to surface
automorphism groups.