Open in CoCalc

KNOT THEORY 101

Cocalc + Sage Worksheet

MAKO 2018

Trefoil

A continuous function whose range is the trefoil knot:

f ⁣:[0,2π]R3f \colon [0, 2\pi] \to \mathbb{R}^3 given by f(t)=((2+cos(3t))cos(2t),  (2+cos(3t))sin(2t),  sin(3t))f(t) = ((2 + \cos(3t))\cos(2t), \ \ (2 + \cos(3t))\sin(2t), \ \ \sin(3t))

3D rendering not yet implemented

Trefoil, again

f ⁣:[0,2π]R3f \colon [0, 2\pi] \to \mathbb{R}^3 given by f(t)=((3+cos(2t))cos(3t), (3+cos(2t))sin(3t), sin(2t))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 times around long way, 3 times around short way.

2 triangles.

tmoebius(2,1,'red',.5) + tknot(2,3)
3D rendering not yet implemented

Torus Knot T(3,2) = Trefoil

3 times around long way, 2 times around short way.

T(3, 2) = T(2, 3).

3 footballs

3D rendering not yet implemented

Torus Knot T(3,4)

3 times around long way, 4 times around short way.

Knot 819, the first nonalternating prime knot.

3 squares.

3D rendering not yet implemented

Torus Knot T(4,3)

3 times around long way, 4 times around short way.

Same as T(3,4)

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, # adaptive = True, 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