Lissajous-toric knots

الأصل · EN

A point in the (N,q)-torus knot in R³ 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((qt+ϕ),(pt+ψ))) while this curve rotates N times around the vertical axis. Such a knot has a natural braid representation Bₙ,q,ₚ which we investigate here. If gcd(q,p)=1, K(N,q,p) is ribbon; if gcd(q,p)=d>1, Bₙ,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.