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.

Braid Groups, by Christian Kassel and Vladimir Turaev
Braids: A Survey, by Joan Birman and Tara Brendle

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. 118-120
2/27	The LKB representation and matrix groups	KT, p. 120-123
3/11	Noodles and spanning arcs	KT, p. 125-136; Bigelow paper	Tell me presentation topics by 3/13
3/13	Noodles, arcs, and second homology	KT, p. 137-147; Bigelow paper
3/18	Faithfulness of LKB; Basics of algorithmic group theory	KT, p. 147-149; BB, p. 60-62
3/20	Garside theory basics; normal forms	KT, p. 256-257; BB, p. 62-64
3/25	Normal forms; conjugacy problem	KT p. 247-249, 254
3/27	Ordering the braid group	Short-Wiest paper, p. 2-8
4/1	Student presentations
4/3	Student presentations
4/8	Student presentations
4/10	Student presentations
4/15	Student presentations
4/17	No class
4/22	Student presentations
4/24	Student presentations
4/29	Student presentations
5/1	Student presentations

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