Spring 2024: Topics in the braid group
This course will serve as a (necessarily eclectic) introduction to the braid group and some of the many places it appears in mathematics. We will begin with the basic theory of the braid group as both the fundamental group of a configuration space and as a mapping class group, and establish elemental structural properties following the treatment in Farb-Margalit. The remainder of the course will be a sequence of largely-independent modules. Background will be presented as necessary. Some possibilities are listed below, but I also welcome suggestions based on the interest of the students.
Here is a Dropbox folder with the lecture notes.
Be aware that these are imperfect and contain some errors and elisions. Always refer to the literature when in doubt.
Below is a tentative list of topics along with references.
- Basics
Seeing the braid group as both the fundamental group of a configuration space and as a mapping class group. Basic structural properties, both algebraic and geometric. Solution to the word problem.- Farb-Margalit, A Primer on Mapping Class Groups, chapter 9
- Birman-Brendle, Braids: a survey (section 5)
- Garside, The braid group and other groups
- ElRifai-Morton, Algorithms for positive braids
- Braids in knot theory
I will present three foundational results (the theorems of Alexander, Markov, and Birman-Menasco) that together assert that every link in the three-sphere admits a (non-unique) expression as the closure of a braid, and describe a simple set of moves that suffice to pass between any two braid representations of the same link. I will also discuss a couple of results about "positivity": Stallings' theorem that positive knots are fibered, and Rudolph's theory of quasipositive braids.- Birman-Brendle, Braids: a survey (sections 2,3)
- Morton, Threading knot diagrams
- Birman-Finkelstein, Studying surfaces via closed braids
- Ghys, A singular mathematical promenade
- Stallings, Constructions of fibred knots and links
- Rudolph, Algebraic functions and closed braids
- Cogolludo, Braid monodromy of algebraic curves
- Representations of braid groups
I will discuss three of the most important representations of the braid group: the representations of Burau, Jones, and Lawrence--Krammer. We will see what each of these tells us about braids and the associated links obtained as their closures.- Birman-Brendle, Braids: a survey (section 4)
- Birman, Braids, Links, and Mapping Class Groups, section 3.3
- Bigelow, The Burau representation is not faithful for n = 5
- McMullen, Braid groups and Hodge theory
- Additional notes on Burau
- Jones, Hecke Algebra Representations of Braid Groups and Link Polynomials
- Bigelow, Braid groups are linear
- Weiyan Chen, Homology of braid groups, the Burau representation, and points on superelliptic curves over finite fields
- Topology of configuration spaces
A closer look at the topology of configuration spaces. Cohomology computations, after Arnol'd and Fuchs. Applications to the inherent complexity of root-finding algorithms, a la Smale.