Posts tagged: knot theory

"The knot is structurally independent of the substrate that carries it.All information in the knot occurs in its relationship with the ambient space.”- Louis Kauffman                                          

Mathematica code:
r[t_] := {Sin[t] + 2 Sin[2*t], Cos[t] - 2 Cos[2*t], -Sin[3*t]};T[t_] := 1/Norm[r'[t]]*r'[t];U[t_] := 1/Norm[r''[t]]*r''[t];V[t_] := Cross[T[t], U[t]];W[a_, d_, t_] := r[t] + d*Cos[a]*U[t] + d*Sin[a]*V[t]Manipulate[With[{d = .5, M = 124, Q = 124}, Graphics3D[  Table[    GraphicsComplex[     Flatten[Table[      W[(a + s)*2 Pi/3, d, t + s*8*Pi/M],     {t, {j*2 Pi/M, (j + 1) 2 Pi/M}}, {a, 0, 2, 1}], 1],      Polygon[{{1, 2, 5, 4}, {2, 3, 6, 5}, {3, 1, 4, 6}}]],  {j, 0, Q, 1}], Lighting -> "Neutral", Boxed -> False, ViewPoint -> Above,  ImageSize -> 600, PlotRange -> 3.5]],{s, 0, 1}]

"The knot is structurally independent of the substrate that carries it.
All information in the knot occurs in its relationship with the ambient space.”
-
Louis Kauffman                                          

Mathematica code:

r[t_] := {Sin[t] + 2 Sin[2*t], Cos[t] - 2 Cos[2*t], -Sin[3*t]};
T[t_] := 1/Norm[r'[t]]*r'[t];
U[t_] := 1/Norm[r''[t]]*r''[t];
V[t_] := Cross[T[t], U[t]];

W[a_, d_, t_] := r[t] + d*Cos[a]*U[t] + d*Sin[a]*V[t]

Manipulate[With[{d = .5, M = 124, Q = 124},
Graphics3D[
Table[
GraphicsComplex[
Flatten[Table[
W[(a + s)*2 Pi/3, d, t + s*8*Pi/M],
{t, {j*2 Pi/M, (j + 1) 2 Pi/M}}, {a, 0, 2, 1}], 1],
Polygon[{{1, 2, 5, 4}, {2, 3, 6, 5}, {3, 1, 4, 6}}]],
{j, 0, Q, 1}],
Lighting -> "Neutral", Boxed -> False, ViewPoint -> Above,
ImageSize -> 600, PlotRange -> 3.5]],
{s, 0, 1}]

nldmut:

Specular holograms by Matthew Brand currently on display at the new Museum of Mathematics in New York.

See his site for more.

The technique used by Brand to create these pieces is not one of conventional holography. He meticulously controls the unique shape of thousands of tiny optical pieces placed on a surface creating a 3D effect when the light source or viewer moves. This is essentially a mathematical problem in differential geometry and combinatorial optimization. Brand was the first person to correctly describe this technique in 2008 even though it dates back as early as the 1930s (check out his paper for details).

The 821 knot parametrized as a Lissajous curve.
Mathematica code:
Manipulate[  Graphics[  Table[    Disk[     {Cos[3 (.1*t + n)*2 Pi/100 + .1], Cos[4 (.1*t + n)*2 Pi/100 + .7]},      .03*Cos[7 (.1*t + n)*2 Pi/100] + .05],   {n, 1, 100}],  PlotRange -> 1.1, ImageSize -> 500],{t,1,10,1}]

The 821 knot parametrized as a Lissajous curve.

Mathematica code:

Manipulate[ 
Graphics[
Table[
Disk[
{Cos[3 (.1*t + n)*2 Pi/100 + .1], Cos[4 (.1*t + n)*2 Pi/100 + .7]},
.03*Cos[7 (.1*t + n)*2 Pi/100] + .05],
{n, 1, 100}],
PlotRange -> 1.1, ImageSize -> 500],
{t,1,10,1}]
Knot Physics

introqft:

Spacetime is assumed to be a branched 4-dimensional manifold embedded in a 6-dimensional Minkowski space. The branches allow quantum interference; each individual branch is a history in the sum-over-histories. A n-manifold embedded in a n+2-space can be knotted. The metric on the spacetime manifold is inherited from the Minkowski space and only allows a particular variety of knots. We show how those knots correspond to the observed particles with corresponding properties. We derive a 2-term Lagrangian. The Lagrangian combined with the geometry of the manifold produces gravity, electromagnetism, weak force, and strong force.


The abstract, topological properties of mathematical knots are related to the properties of elementary particles in the physical universe.

The interested reader can view the paper as a .pdf here.

(via introqft)

A 3D orthographic projection of a 4D slice knot rotating while keeping two orthogonal planes fixed.

Created with KnotPlot

more


A 3D perspective projection of a 4D slice knot rotating while keeping two orthogonal planes fixed.

Created with KnotPlot

more

Louis Kauffman’s notes on Fox calculus. I don’t really get it.



This is a shiny torus knot.



Lynnclaire Dennis had a near-death experience and brought back some interesting geometric ideas including the Pattern knot, which is shown here rotating on a torus.

Infinite Whitehead Tangle

This never-ending knot, animated by Kenneth Baker, was made by concatenating a sequence of smaller tangles that look like this:

A quarter-turn is applied when passing from one copy of this tangle to the next in the infinite sequence.  Here’s another view:

Sketches of Topology is an amazing blog for visuals like these, and I will probably crib from it frequently.

(via hearseethink)

The trefoil knot is the simplest example of a nontrivial knot, meaning that it is impossible to untie the knot in three dimensions without cutting it. The trefoil knot can be defined as the curve given by the parametric equations
$$ x=(2+cos3t)cos2t ,$$$$ y=(2+cos3t)sin2t ,$$$$ z=sin3t .$$

The trefoil knot is the simplest example of a nontrivial knot, meaning that it is impossible to untie the knot in three dimensions without cutting it. The trefoil knot can be defined as the curve given by the parametric equations

$$ x=(2+cos3t)cos2t ,$$
$$ y=(2+cos3t)sin2t ,$$
$$ z=sin3t .$$