How to undo a type I Reidemeister move only using type II and type III moves?

Question

I am pretty new to geometric topology and I am struggling to solve this problem. I have tried to approach this a lot of different ways and I will spare you the drawings. I am trying to undo this portion of a link that you would use type I moves for, but only using type II and type II moves.

Answer 1

Type I moves change the writhe of a link diagram by $\pm 1$, where type II and type III moves leave writhe invariant. Restricting your moves to types II and III is called regular isotopy (though for diagrams in $\mathbb{R}^2$ there is usually a type I' move where you insert a canceling pair of type I moves), which is more restrictive.

It might be helpful if you give a diagram of the link in question.

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With the diagram you've now posted in mind, this is a case where the writhes cancel and there is a sequence of moves of types II and III. See for example https://mathoverflow.net/a/162729/43804, but note that they used II followed by III to get from the first to second diagram. This figure shows how you can think of this as sort of untwisting out of a thick rope: https://www.researchgate.net/figure/The-Whitney-trick_fig9_47820408 (you can think of the reverse as taking a portion of a long rope then twisting that portion about an axis perpendicular to the diagram).

I would suggest finding some thick rope (which resists changes in writhe, so therefore type I moves) and see how it helps you obtain a move sequence.