Ultraknots and limit knots

Abstract

We prove that every knot type in ℝ³ can be parametrised by a smooth function 𝑓 : 𝑆¹ → ℝ³, 𝑓(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) such that all derivatives 𝑓^(𝑛)(𝑡) = ( 𝑥^(𝑛)(𝑡), 𝑦^(𝑛)(𝑡), 𝑧^(𝑛)(𝑡) ), 𝑛 ∈ ℕ, parametrise knots and every knot type appears in the corresponding sequence of knots. We also study knot types that arise as limits of such sequences.