Trivial knot with some crossing points

Question

Suppose $K$ is a knot diagram of a trivial knot with some crossing points. Then we know that there is a finite sequence of Reidemeister moves to transform $K$ to a trivial diagram (diagram with no crossings). Now I’m asking about the sequence ? Is there an example of such $K$ such that the Reidemeister moves which increase the number of crossing points must be involved in any sequence transforming $K$ to trivial? In other words, can we say that any sequence to eliminate the crossing points does not involve creation of crossing points? I know this is not true in general ( transform a diagram to a diagram with minimal number of crossings) but I wonder if it’s true for trivial diagram

Answer 1

There is a diagram of the unknot such that any sequence of Reidemeister moves transforming the diagram into the standard diagram must increase the number of crossings. Here is an example due to Kauffman and Lambropoulou:

One can prove that the number of crossings must increase in any sequence by performing all available Reidemeister 3 moves and observing that there are no Reidemeister 1 or 2 moves available that decrease the number of crossings. An unknotting sequence for the diagram is below (also taken from the paper of Kauffman and Lambropoulou).

See this related discussion on MathOverflow.