MATH 7590: Geometric Topology Homework

MATH 7590   Geometric Topology

Fall 2003

Some Homework Problems & Potential Presentation Topics


  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 Bn is generated by the elementary braid sigma1 and
    the braid eta = sigma1*sigma2*...*sigman-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 FG(C*,n) and FG(C*,n)/Sn,
    where G is a finite cylic group and Sn is the symmetric group.

Dan Cohen                           Fall 2003
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