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Math 9100: Braid Groups

Spring 2025

Meets: Tue/Thu 2:00 - 3:20 in Wachman Hall, room 527
Instructor: David Futer
Office: 1026 Wachman Hall
Office Hours: Tue 10:00 - 12:00, Wed 11:00 - 12:00, or by appointment
E-mail: dfuter at temple.edu
Phone: (215) 204-7854

Course content: This course will be an introduction to the theory of braid groups. We will discuss how braids arise in knot theory, algebraic geometry, the study of configuration spaces, and more. From there, we can proceed in various directions, guided by the interests of the class. Possibilities include:

  • Geometric group theory of the braid group (residual finiteness, incoherence, orderability, ...)
  • Representations of the braid group (Burau, Lawrence-Krammer-Bigelow)
  • Connections between braid representations and knot invariants
  • Connections to quantum topology
  • Garside structures and algorithms, automatic structures

References: We will primarily work from the following sources.

These will be supplemented with occasional readings from more recent papers.

Prerequisites: Math 8011-12 and 8061-62.

Grading:

  • 60% of the grade will be based on homework.
  • 40% of the grade will be based on an in-class presentation.

Detailed schedule

This will be gradually filled in as the semester progresses. BB = Birman-Brendle; KT = Kassel-Turaev.

Day Topic Reading Homework/Note
1/14 Braid diagrams, configuration spaces, generators & relations BB, p. 3-5, KT, p. 4-17.
1/16 Braids and mapping class groups BB, p. 5-8; KT, p. 35-40
1/21 Braid automorphisms of free groups KT, p. 31-34
1/23 Homomorphisms between braid groups KT, p. 18-20, 103-105 Homework 1, due 1/30
1/28 Torsion-freeness, residual finiteness Stallings paper
1/30 Center and abelianization of the braid group KT, p. 20-24
2/4 Links and closed braids KT, p. 47-50, 58-60
2/6 Isotopy of links and conjugacy of braids KT, p. 54-57
2/11 The Burau representation and homology KT, p. 93-100.
2/13 Non-faithfulness of Burau representation KT, p. 100-107
2/18 Reduced Burau rep; faithfulness of ψ3 KT, p. 107-111; Formanek paper Homework 2, due 2/27
2/20 Alexander-Conway polynomial KT, p. 111-118
2/25 Lawrence-Krammer Bigelow representation KT, p. 118-123
2/27 Noodles and spanning arcs KT, p. 124-128

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Last modified: Fri Aug 21 13:41:22 PDT 2009