Reidemeister infinity move

Question

If you have a knot K. How do you apply a R infinity move to it? Is their an algorithm which tells you what to do at certain parts on a knot?

Answer 1

To visualise R(infinity) imagine a knot diagram on the surface of a sphere. Take a simple arc on the bottom edge of our diagram and run it around the back of the sphere so that it is sitting on top of our diagram. It forms a kind of skipping rope motion around the body of the knot. This operation is also valid in $\mathbb{R}^2$ since $\mathbb{R}^2$ is homeomorphic to $\mathbb{S}^2$ with one point removed.

"Is there an algorithm that tells you what to do?" I guess it depends what you want to do! The main use I have seen for R(infinity) is for giving an oriented link a closed braid representation (Alexander). Informally the way we do this is by repeated application of R(infinity) and the other Reidermeister moves, so that the link is close to a collection of concentric circles. Once the Seifert circles are concentric and with the same orientation your link is a closed braid.