Algebraic Family of Trefoil Curves

display

surface intersection

equatorial cross section

knot curve only

period function

shape parameter, k

0.5

H

1

= -0.5

3

X

+

2

X

Z+1.5X

2

Y

+

2

Y

Z-0.5625Z = 0

H

2

=

4

X

+2

2

X

2

Y

-6

2

X

Y+0.75

2

X

+

4

Y

+2

3

Y

+0.75

2

Y

-3.

2

Z

= 0

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,

V

T

(k)=(x,y,z)∈

3



:

H

1

(k;x,y,z)=0,

H

2

(k;x,y,z)=0

(the intersection of two surfaces), defines a natural time parameter

t

relative to the tangent geometry. It is then possible to integrate the period function

T(k)=∮dt

by solving a second-order ordinary differential equation in the shape variable

k

(see Details).