  A Lissajous knot is one that can be parameterized by a single cosine function
in each coordinate. Lissajous knots are highly symmetric, and for this reason,
not all knots are Lissajous. We prove several theorems which allow us to place
bounds on the number of Lissajous knot types with given frequencies and to
efficiently sample all possible Lissajous knots with a given set of
frequencies. In particular, we systematically tabulate all Lissajous knots with
small frequencies and as a result substantially enlarge the tables of known
Lissajous knots.
  A Fourier (i, j, k) knot is similar to a Lissajous knot except that each
coordinate is now described by a finite sum of i, j, and k cosine functions
respectively. According to Lamm, every knot is a Fourier-(1,1,k) knot for some
k. By randomly searching the set of Fourier-(1,1,2) knots we find that all
2-bridge knots up to 14 crossings are either Lissajous or Fourier-(1,1,2)
knots. We show that all twist knots are Fourier-(1,1,2) knots and give evidence
suggesting that all torus knots are Fourier-(1,1,2) knots.
  As a result of our computer search, several knots with relatively small
crossing numbers are identified as potential counterexamples to interesting
conjectures.
