Problems

1. Under what conditions on a braid is the closure a knot?
   
2. Identify the kernel of the map from the 3-string braid group B₃ to SL₂(Z) given in class.
   
3. Show that the n-string braid group B_n is generated by the elementary braid sigma₁ and
   the braid eta = sigma₁*sigma₂*...*sigma_n-1.
4. Verify that the relations given in class for the pure braid group hold, either geometrically
   or using the Artin representation (or both).

Potential Presentation Topics

1. Study the braid groups of (orientable) surfaces.
   E.g., find presentations; discuss the Dirac string problem...
2. Use the Reidemeister-Schreier rewriting process to obtain a presentation for the (abstract)
   pure braid group from the homomorphism from the (abstract) full braid group to the symmetric group.
3. Write computer programs to implement the Artin representation; the Dehornoy reduction algorithm.
4. Investigate the structure of the fundamental groups of the orbit configuration spaces F_G(C^*,n) and F_G(C^*,n)/S_n,
   where G is a finite cylic group and S_n is the symmetric group.