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.
References:
- Braid Groups, by Christian Kassel and Vladimir Turaev
- Braids: A Survey, by Joan Birman and Tara Brendle
Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.
Detailed schedule
| 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 groups | KT, 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 |
| 10/28 | Garside theory basics | KT, p. 256-257 | |
| 10/30 | Normal 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 |
dfuter at temple edu Last modified: Fri Aug 21 13:41:22 PDT 2009