@RegularGraph his point is valid, but hard to understand. His picture convinced me. I commented on it in my update of the question.

@RegularGraph I did not edit anything out of the regular post. I only made updates.

@Ryanbudney the counterexample given in these comments appears to me to be impossible to undo given any material. I have tried it on magician’s rope.

If you read my update, my proposal did not work.

My experience is that with magician’s rope, my proposal would probably work. It is both flexible and sturdy. I will try it out on the hard unknots that were listed in the links you gave.

yes thank you very much for your help and the link. I am new to the subject matter of knots, except for my experience with them in magic.

All I am saying is that in real life, it is not so difficult to recognize an unknot. Thus, if a computer simulates real life, it shouldn’t be so difficult to recognize an unknot on a computer.

your assessment is correct, but I don’t think this is a bad thing. This is how problems get solved.

@ryanbudney I’m not sure what you mean. I read your comment before. The physical attributes of the string might affect things in the real world but on a computer they can be easily adjusted. We might as well make it frictionless.

if the problem is the length of sticks as shown in the first paper (or even the width of the sticks), whenever the algorithm runs into this issue, it could bypass it by making the sticks longer and thinner or add breakpoints if that doesn’t work. If it continues to do this, it will get the string unknotted. And probably polynomial time too since this little adjustment shouldn’t take too long.

These papers give problems that such an algorithm may encounter (unknots that cannot be unknotted) but also give solutions on a computer (change the lengths of the sticks so they can be unknotted). They might be problematic in the physical world but not in the virtual world.

@AndyPutman I updated my question. I still think the question is a good question even though my definition of unknot is not so rigorous.

@AndyPutman now I understand. Many magic tricks with rope are built around that principle.

@AndyPutman sorry about the name misspelling. Am I understanding you correctly that this problem may have a different answer for a rope that has two ends with a restriction on what moves can be made to unknot it than a rope in which the ends are glued together?