2016/10/14 by Marc Soret, Soret, Marc, Marina Ville +1
Mathematics · #57M25 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.1610.04418
31 pages, 11 figures
openalex publication_date 2016/10/14 · arxiv created 2016/10/30 · arxiv updated 2016/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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↦(sin(qt+ϕ),cos(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.