text: abstract. we solve the simultaneous conjugacy problem in artin’s braid groups and,
  more generally, in garside groups, by means of a complete, effectively computable, finite
  invariant. this invariant generalizes the one-dimensional notion of super summit set to
  arbitrary dimensions. one key ingredient in our solution is the introduction of a provable
  high-dimensional version of the birman–ko–lee cycling theorem. the complexity of this
  solution is a small degree polynomial in the cardinalities of our generalized super summit
  sets and the input parameters. computer experiments suggest that the cardinality of this
  invariant, for a list of order n independent elements of artin’s braid group bn , is generically
  close to 1.