Math 7590: Braid Portfolio
Proposal & First Review: Thursday, 4 March.
By end of mid-term week, turn in one written review & your selected
presentation topic with a brief summary and initial list of references
20 Minute Presentation: Week following Spring Break: April 14, 16 & 21
Portfolio: Final Exam: 12 Noon, Sat., May 9
Electronic Version of Project, Reviews, Exercise (All files needed for
total reconstruction: LaTeX, Figures, Source Code, PDF) in one .gz(ip)
(Mac/Linux) or ZIP (Windows) archive and attached via email.
Grades: Grading will be weighted as follows:
Project: (see below) 50%
Research Paper Reviews: 20%
Other "portfolio" items: 30%
Worked Exercises or Examples
Revised & Expanded Lecture Notes for a single topic
Normal Forms: Thurston, Artin, Handle Reduction, Garside
Growth Functions for Braid/Garside Groups (see Dehornoy/Weist)
Review/Summary of Research Article:
In the style of a Math Review: http://www.ams.org/mathscinet
write three one-page summaries of published articles on braids.
Birman Bibliograph: see Course Portfolio Web Page
Braids: A Survey by Birman, Joan S. & Brendle, Tara E.
Open Problem Lists: See appropriate sections of the lists in the Arkiv
Benson Farb book: Problems on Mapping Class Groups
math.GT/0406190 Problems on invariants of knots and 3-manifolds.
Other On-Line Courses on Braids:
See references in "Possible Project Topics" link
Project: Develop a topic related to braids. An oral presentation of your
project will be made after Spring Break. A written report will be submitted
with the portfolio by the Final Exam date. (Precise dates above.)
Jones Polynomial and Yang Baxter Equation for Quantum sl_2:
q-alg/9506002 Title: Links, Quantum Groups, and TQFT's
Authors: Stephen Sawin
Abstract: The Jones polynomial and the Kauffman bracket are constructed, and
their relation with knot and link theory is described. The quantum groups and
tangle functor formalisms for understanding these invariants and their
descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the
Jones polynomial, is constructed explicitly.
Representations of Braids
Non-injectivity of the Burau Representation: Moody/Long/Bigelow
Reidemeister-Schreier Proof of Pure Braid Presentation
Thompson Group: Greenberg/Sergiescu Comm. Math. Helv. vol. 66
Artin Groups/Garside Groups: Papers of Patrick Dehornoy & Ruth Charney
Algebras Related to Braid Theory:
Cord Algebra: math/0407071 Framed knot contact homology. Lenhard Ng
Braids detected by Finite Type Invariants: Hutchings/Bar Natan/
Quandles & Braids:
M Eisermann: Yang-Baxter deformations of quandles and racks
Braid Ordering Algorithm: Dehornoy/Weist
Free Differential Calculus: Group Cohomology of Semi-Direct Products
Applications to Physics: Braids, Yang-Baxter & Quantum Groups
Statistical Mechanics: Subfactors and Knots (CBMS Regional Conference
Series in Mathematics) by Vaughan F. R. Jones
Lou Kauffman: Book: Knots & Physics Section 1.8, 1.10
DNA: Dewitt L. Sumners
Protein Folding: [ps] arXiv:0902.1025 Fatgraph Models of
Braid Group Cryptography. arXiv:0711.3941 David Garber.
Braids & Juggling: Satyan L. Devadoss math.GT/0602476
Visualization of Seifert Surfaces or Turaev Surfaces
Further Ingredients for a Mathematical Portfolio
Help Improve Mathematics on the Internet
How correct any mathematical errors in the material on braid theory on these Internet sites!
Solve an Open Problems