Lissajous-toric knots

Abstract

A point in the [Formula: see text]-torus knot in [Formula: see text] goes [Formula: see text] times along a vertical circle while this circle rotates [Formula: see text] times around the vertical axis. In the Lissajous-toric knot [Formula: see text], the point goes along a vertical Lissajous curve (parametrized by [Formula: see text] while this curve rotates [Formula: see text] times around the vertical axis. Such a knot has a natural braid representation [Formula: see text] which we investigate here. If [Formula: see text], [Formula: see text] is ribbon; if [Formula: see text], [Formula: see text] is the [Formula: see text]th power of a braid which closes in a ribbon knot. We give an upper bound for the [Formula: see text]-genus of [Formula: see text] in the spirit of the genus of torus knots; we also give examples of [Formula: see text]’s which are trivial knots.