Is there any knot showing infinitely many crossings?

Question

Let $K$ be any unknot. Is it possible that $K$ shows infinitely many crossings?

And if it is possible: How to get $K$ from the simplest unknot through Reidemeister moves?

Answer 1

There are wild knots with infinitely many crossings. A knot is wild if it is not tame, where this means that the knot extends to an embedded $S^1 \times D^2 \subset \mathbb{R}^3$. Every tame knot can be realized as a finite polygonal path.

Note that a smooth or PL knot is necessarily tame.

If you take the idea for the construction of the wild knot in the image in that wikipedia article but do it with different crossings you can make it an unknot.

Knot diagrams for equivalent tame knots have a finite sequence of Reidemeister moves relating them. This doesn't work for wild knots.

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A famous use of wild knots is the Mazur swindle.