is a symmetry of the Lissajous knot. In general, a knot that has a symmetry of this type is called 2-periodic, so every even Lissajous knot must be 2-periodic. Consequences The symmetry of a Lissajous knot puts severe constraints on the Alexander polynomial. In the odd case, the Alexander polynomial of the Lissajous knot must be a perfect square. In the even case, the Alexander polynomial must be a perfect square modulo 2. In addition, the Arf invariant of a Lissajous knot must be zero. It follows that: The trefoil knot and figure-eight knot are not Lissajous. No torus knot can be Lissajous. No fibered 2-bridge knot can be Lissajous. References Knots (knot theory)