  Murasugi discovered two criteria that must be satisfied by the Alexander
polynomial of a periodic knot. We generalize these to the case of twisted
Alexander polynomials. Examples demonstrate the application of these new
criteria, including to knots with trivial Alexander polynomial, such as the two
polynomial 1 knots with 11 crossings.
  Hartley found a restrictive condition satisfied by the Alexander polynomial
of any freely periodic knot. We generalize this result to the twisted Alexander
polynomial and illustrate the applicability of this extension in cases in which
Hartley's criterion does not apply.
