See C. Lamm:
Deformation of cylinder knots, 4th chapter of Ph.D. thesis, ‘Zylinder-Knoten und symmetrische Vereinigungen‘, Bonner Mathematische Schriften 321 (1999), available since 2012 as arXiv:1210.6639.
See E. Kin, H. Nakamura and H. Ogawa: Lissajous 3-braids, J. Math. Soc. Japan 75, 195--228 (2023) (or arXiv:2008.00585v4).