A family of pseudo-Anosov braids with large conjugacy invariant sets
  We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge. 
Byung Hee An and Ki Hyoung Ko
math.GT math.GR