Math 7590 Portfolio Information

Portfolio Ideas

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
Algorithmic Development:  

	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:
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 
		Robion Kirby
		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

Birman-Murakami-Wenzl Algebra

Braids detected by Finite Type Invariants:  Hutchings/Bar Natan/

Quandles & Braids:  

	M Eisermann:  Yang-Baxter deformations of quandles and racks

Algorithm Development

	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

Other Applications:

	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!
  • Wolfram MathWorld
  • Wikipedia
  • Daily Notes

  • Polish and Illustrate (using LaTeX), a week of notes.
  • Algorithm Implementation

  • Braids in Bar-Natan's KnotTheory
  • The Knot Atlas: Bar-Natan et al
  • Stephen Bigelow: Braid Applet
  • Wolfram Demonstrations: Knots: Live App
  • Early Sources

  • Birman's Early Bibliography (see book)
  • Projects

    Solve an Open Problems

  • Birman's Open Problems (see book)