Math 801: Topics in Applied Mathematics
Braids (Spring 2008)
All lectures in
(djvu format is much smaller)
Lecture 2: Definitions
Lecture 3: Artin
Lecture 4: Fundamental Groups.
Lecture 5: Configuration
Lecture 6: The Presentation
The Presentation Theorem II: The Pure Braid Group.
Lecture 9: The Dirac
String Trick. [
10–12: Mapping Class Groups.
Mapping Class Groups of General Surfaces. [incomplete]
Lectures 13–14: The
Mapping Class Group of the Torus.
Lectures 15–16: The
Singularities of Foliations.
Representations of Bn.
Burau and Homology.
Entropy and the Fundamental Group.
Action on π1(M) for the Torus; Manning's Theorem.
Subshifts of Finite Type.
Entropy of pseudo-Anosov Diffeomorphisms.
Markov Partition for pseudo-Anosovs.
From Markov Partitions to Train Tracks.
Train Track Graphs.
Normal Train Tracks and Folding.
Measured Train Tracks and Fibered Neighbouroods.
Train Track Automata.
Train Track Automata, part II: D4 and Culs-de-sac.
Minimising the Dilatation.
Maximising the Dilatation.
Computer Implementation of Train Track Automata.
Bibliography and Resources
- J. S. Birman, Braids, Links and Mapping Class Groups, Annals of
Mathematical Studies 82, Princeton University Press,
- V. L. Hansen, Braids and Coverings, London Mathematical Society
Student Texts 18, Cambridge University Press, 1989.
- Denis Auroux's
- D. Rolfsen, "New
developments in the theory of Artin's braid groups," Topology and
its Applications 127, 2003.
- J. S. Birman and T. E. Brendle, "Braids: A Survey,"
- M. Epple, "Orbits of asteroids, a braid,
and the first link invariant," Mathematical Intelligencer 20, 45,
- E. Artin, "Theory of braids," Annals of Mathematics 48, 101,
- B. Farb and D. Margalit, A Primer on Mapping
Class Groups, version 2.95, August 2007.
- Lee Mosher's web
site has several long works on mapping class groups. See also his
Notices article "What is a
- W. P. Thurston, The Geometry and
Topology of Three-Manifolds, Electronic version 1.1 — March
- J. Milnor,
Foliations and Foliated Vector Bundles, MIT lecture notes,
- P. L. Boyland, "Isotopy Stability of Dynamics
on Surfaces," 1999.
- P. L. Boyland and J. Franks, Notes
on Dynamics of Surface Homeomorphisms, University of Warwick,
- G. Band and P. L. Boyland, "The Burau estimate for the
entropy of a braid," 2006.
- A. Fathi, F. Laundenbach, and V. Poénaru, Travaux de
Thurston sur les surfaces, Astérisque 66–67,
- A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after
Nielsen and Thurston, London Mathematical Society Student
Texts 9, Cambridge University Press, 1988.
- J.-Y. Ham and W. T. Song, "The
Minimum Dilatation of Pseudo-Anosov 5-Braids,"
Experiment. Math. 16, 167–180, 2007.
- R. C. Penner and J. L. Harer, Combinatorics of Train
Tracks, Annals of Mathematical Studies 125, Princeton
University Press, 1992.
Thurston's famous cartoon of a train track.