Math 9100: Braid Groups

Fall 2019

Meets: Mon/Wed 10:30 - 11:50 in Wachman Hall, room 527
Instructor: David Futer
Office: 1026 Wachman Hall
Office Hours: Mon/Tue 1:00 - 2:30, 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. We will study two well-known linear representations of braid groups, as well as the connection that these representations have to knot invariants. We will also discuss Garside structures on braids and their connections to algorithmic problems in braid groups. Finally, we will discuss orders on braid groups.


Prerequisites: Math 8011-12 and 8061-62.

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.

Detailed schedule

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

Day Topic Reading Homework/Note
8/26 Braid groups and configuration spaces BB, p. 3-5.
8/28 Braids and braid diagrams KT, p. 4-17.
9/4 Braids and mapping class groups BB, p. 5-8; KT, p. 31-44 Homework 1, due 9/11
9/9 Homomorphisms between braid groups KT, p. 18-12
9/11 Residual finiteness Stallings paper
9/16 Birman exact sequence, SES of pure braid groupsKT, p. 21-22 & 27-29
9/18 Center and abelianization of the braid group KT, p. 22-24
9/23 Links and closed braids KT, p. 47-57
9/25 The Burau representation KT, p. 93-95, 107-109. Homework 2, due 10/2
9/30 Reduced Burau rep; homological interpretation KT, p. 98-100, 107-110
10/2 Non-faithfulness of Burau representation KT, p. 100-107
10/7 Alexander-Conway polynomial KT, p. 111-118
10/9 No class
10/14 Lawrence-Krammer Bigelow representation KT, p. 118-123
10/16 Noodles and spanning arcs KT, p. 124-128
10/21 Faithfulness of LKB KT, p. 128-149; Bigelow paper
10/23 Word and conjugacy problems BB, p. 60-62; Wikipedia Homework 3, due 10/30
10/28Garside theory basics KT, p. 256-257
10/30Normal forms and algorithms KT p. 247-249, 254
11/4 Least common divisors; torsion-freeness KT p. 258-259
11/6 Conjugacy via super summit sets and cycling BB p. 64-66; ElRifai-Morton paper
11/11 Ordering the braid group Short-Wiest paper, p. 2-8
11/13 Automatic groups Epstein et al, Chapter 2
11/18 Automatic structures on braid grops Epstein et al, p. 190-201
11/20 Regular languages and growth Flajolet-Sedgewick, Section V.5; DFW paper
12/2 Presentation: Delaney
12/4 Presentation: Katherine
12/9 Presentation: Rosie
12/11 Presentations: DB and Ruth

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