Event № 256
Joint work with Marc Soret. In a (N,q)-torus knot, a particle goes q times around a vertical planar circle which is being rotated N times around a central axis. On a Lissajous toric knot K(N,q,p), the particle goes through a Lissajous curve parametrized by (sin(qt), cos(pt+u)) while we rotate this curve N times around a central axis; we assume (N,q)=(N,p)=1. Christopher Lamm first defined these knots as billiard knots in the solid torus and we encountered them as singularity knots of minimal surfaces in R^4. They are naturally presented as closed braids which we write precisely: we derive that they are all ribbon or periodic, as stated by Lamm. Finally we give an upper bound for the 4-genus of K(N,q,p) in the spirit of the 4-genus of the torus knot.