Algebraic Family of Trefoil Curves
Algebraic Family of Trefoil Curves
The trefoil is the simplest nontrivial knot and the only knot with three crossings [1]. In this Demonstration the trefoil is drawn with cyclic symmetry using the parametric equations
x(ϕ)=ksinϕ+sin(2ϕ)
y(ϕ)=kcosϕ-cos(2ϕ)
z(ϕ)=sin(3ϕ)
with .
k∈(-1,1)\{0}
An encoding of the knotted curve as an algebraic variety, (k)=(x,y,z)∈:(k;x,y,z)=0,(k;x,y,z)=0 (the intersection of two surfaces), defines a natural time parameter relative to the tangent geometry. It is then possible to integrate the period function by solving a second-order ordinary differential equation in the shape variable (see Details).
V
T
3
H
1
H
2
t
T(k)=∮dt
k