Lissajous-toric knots

Marc Soret (1) , Marina Ville (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

A 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.

Mots clés: Braid presentations; knots in S 3; minimal surfaces; four-genus; torus knot; Lissajous curve; ribbon knot

Domaines: Géométrie différentielle [math.DG]

Marc Soret, Marina Ville. Lissajous-toric knots. Journal of Knot Theory and Its Ramifications, 2020, 29 (01), pp.2050003. ⟨10.1142/S0218216520500030⟩. ⟨hal-03064345⟩