Lissajous-toric knots.

A point in the (N , q) -torus knot in ℝ 3 goes q times along a vertical circle while this circle rotates N times around the vertical axis. In the Lissajous-toric knot K (N , q , p) , the point goes along a vertical Lissajous curve (parametrized by t ↦ (sin (q t + ϕ) , cos (p t + ψ))) while this curve rotates N times around the vertical axis. Such a knot has a natural braid representation B N , q , p which we investigate here. If gcd (q , p) = 1 , K (N , q , p) is ribbon; if gcd (q , p) = d > 1 , B N , q , p is the d th power of a braid which closes in a ribbon knot. We give an upper bound for the 4 -genus of K (N , q , p) in the spirit of the genus of torus knots; we also give examples of K (N , q , p) 's which are trivial knots. [ABSTRACT FROM AUTHOR]