No knots in 4D


Exhibit: table of content

Mathematics Math21a, Spring 2006
Multivariable Calculus
Oliver Knill, SciCtr 434, knill.harvard.edu

Unknotting a knot in 4D



unknotting a knot A knot is a closed curve in space. A knot is called trivial, if one can deform it to a simple unknotted circle without having any selfintersections at any time. It is quite easy to see that in four dimensions, there are no nontrivial knots. You would not be able to tie a shoe in four dimensional space.

We use color as the fourth coordinate. The fourth dimension is the "Hue value" between 0 and 1. It labels the colors similar to the rainbow. f we color the knot, we place the knot in four dimensional space. We have a parametrized curve r(t) = (x(t),y(t),z(t),c(t)). The three first positions (x,y,z) are the positions of the point on the curve and the four position c is the color at that point. If we unknot the knot, we can now do that in four dimensions as long as the colors are differents, where the projection of the curve to three dimensions intersects


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Oliver Knill, Math21a, Multivariable Calculus, Spring 2006, Department of Mathematics, Faculty of Art and Sciences, Harvard University