Parametrization of a knot isotopy

Question

I am working on a computer visualization of a knot isotopy of the standard unknot embedding to an unknot with a Reidemeister I move. Does anyone have a formula?

Answer 1

This is very likely not what you're looking for, but I think it illuminates what a Reidemeister I move "is".

Consider this parameterization of an unknotted circle:

Depending on which direction you're looking at it, the projected diagram has either zero or one crossings:

So, by animating the projections of this loop as we rotate it, we get this animation illustrating a sequence of Reidemeister I moves:

The homotopy of these plane curves is $$f_t(\theta) = (\cos t \sin 2\theta + \sin t \cos \theta, \sin\theta).$$ (Note: in the above animation the slider is mislabeled and it should read $t$ rather than $\theta$.)

The Reidemeister moves come from examining the kinds of singularities in diagrams that occur generically during an isotopy of the projected knots, and Reidemeister I moves come, basically, from when a portion of a curve like this rotates.

The following is potentially more useful. If you analyze a Reidemeister I singularity carefully, you can isotope the isotopy to give a nice local model for it. This is a polynomially defined local model for the Reidemeister I singularity: $$f_t(s)=(s^2,s,s^3-ts)$$ where $s\in\mathbb{R}$ and $t\in[-1,1]$.

Projecting this onto the XZ axis gives this:

Projecting it onto the YZ plane, we can see that what's happening is that a bit of the knot is being turned over: