Open in CoCalc

# KNOT THEORY 101

## Trefoil

A continuous function whose range is the trefoil knot:

$f \colon [0, 2\pi] \to \mathbb{R}^3$ given by $f(t) = ((2 + \cos(3t))\cos(2t), \ \ (2 + \cos(3t))\sin(2t), \ \ \sin(3t))$

3D rendering not yet implemented

## Trefoil, again

$f \colon [0, 2\pi] \to \mathbb{R}^3$ given by $f(t) = \left((3 + \cos(2t))\cos(3t), \ (3 + \cos(2t))\sin(3t), \ \sin(2t)\right)$

3D rendering not yet implemented

## Figure-Eight Knot

def f(t): return ((2 + 1*cos(2*t))*cos(3*t), (2 + 1*cos(2*t))*sin(3*t), 1*sin(4*t));
show(
parametric_plot3d(
f(t),
(0, 2*pi),
frame = false,
plot_points = 300,
color='black',
thickness = 6
))

3D rendering not yet implemented

## Torus Knot T(2,3) = Trefoil

#### 2 triangles.

tmoebius(2,1,'red',.5) + tknot(2,3)

3D rendering not yet implemented

## Torus Knot T(3,2) = Trefoil

#### 3 footballs

3D rendering not yet implemented

## Torus Knot T(3,4)

#### 3 squares.

3D rendering not yet implemented

## Torus Knot T(4,3)

#### 4 triangles.

tmoebius(2,1,'red',.5) + (lambda bigr, littler: parametric_plot3d(
((bigr + cos(littler*t))*cos(bigr*t),
(bigr + cos(littler*t))*sin(bigr*t),
sin(littler*t)),
(t, 0, 4*pi),
frame = false,
plot_points = 300,
color='black',
thickness = 4
))(2,1.5)

3D rendering not yet implemented

## T(11,12), crossing number 120

(lambda bigr, littler: parametric_plot3d(
((bigr + cos(littler*t))*cos(bigr*t),
(bigr + cos(littler*t))*sin(bigr*t),
sin(littler*t)),
(t, 0, 10*pi),
frame = false,
plot_points = 300,
color='black',
thickness = 4
)
+    parametric_plot3d(
((bigr + cos(v))*cos(u),
(bigr + cos(v))*sin(u),
sin(v)),
(u, 0, 2*pi),
(v, 0, 2*pi),
color='red',
opacity = 0.0,
frame = false))(2.2,2.4)

3D rendering not yet implemented

## 1-Twist Moebius Strip

moebius(2,.5,1,'blue',.5)

3D rendering not yet implemented

## 2-Twist Moebius Strip

moebius(2,.5,2,'blue',.5)

3D rendering not yet implemented

## 3-Twist Moebius Strip

3D rendering not yet implemented

## Torus Knot T(2,3) again, with Moebius Strip

3D rendering not yet implemented

## Same, with a torus

3D rendering not yet implemented

## Torus Knot T(2,5) on a 5-Twist Moebius Strip

3D rendering not yet implemented

## Torus Knot T(2,11) with an 11-Twist Moebius Strip

moebius(1,.5,11,'red',.8) + mknot(1,.5,11)

3D rendering not yet implemented