Lissajous-toric knots

International audienceA 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(qt + φ), cos(pt + 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 dth 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