
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 
Email:  dfuter
at
temple.edu 
Phone:  (215)
2047854 
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 wellknown 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:
Prerequisites: Math 801112 and 806162.
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 =
BirmanBrendle; KT = KasselTuraev.
Day 
Topic 
Reading 
Homework/Note 
8/26  Braid groups and configuration spaces  BB, p. 35. 
 8/28  Braids and braid diagrams  KT, p. 417. 
 9/4  Braids and mapping class groups  BB, p. 58; KT,
p. 3144  Homework 1, due 9/11
 9/9  Homomorphisms between braid groups  KT, p. 1812 
 9/11  Residual finiteness  Stallings paper 
 9/16  Birman exact sequence, SES of pure braid groups  KT,
p. 2122 & 2729 
 9/18  Center and abelianization of the braid group  KT, p. 2224 
 9/23  Links and closed braids  KT, p. 4757 
 9/25  The Burau representation  KT, p. 9395,
107109.  Homework 2, due 10/2
 9/30  Reduced Burau rep; homological interpretation  KT,
p. 98100, 107110 
 10/2  Nonfaithfulness of Burau representation  KT, p. 100107 
 10/7  AlexanderConway polynomial  KT, p. 111118 
 10/9  No class  
 10/14  LawrenceKrammer Bigelow representation  KT,
p. 118123 
 10/16  Noodles and spanning arcs  KT, p. 124128 
 10/21  Faithfulness of LKB  KT,
p. 128149; Bigelow
paper 
 10/23  Word and conjugacy problems  BB,
p. 6062; Wikipedia 
Homework 3, due 10/30
 10/28  Garside theory basics  KT, p. 256257 
 10/30  Normal forms and algorithms  KT p. 247249, 254 
 11/4  Least common divisors; torsionfreeness 
KT p. 258259 
 11/6  Conjugacy via super summit sets and
cycling  BB p. 6466; ElRifaiMorton
paper 
 11/11  Ordering the braid
group  ShortWiest
paper, p. 28 
 11/13  Automatic groups  Epstein et al, Chapter 2 
 11/18  Automatic structures on braid
grops  Epstein et al, p. 190201 
 11/20  Regular languages and
growth  FlajoletSedgewick,
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
