This is just a comment. The same week (!) when Dylan asked this question, we received at our department a message from a non-professional mathematician who wrote a computer program that tries to simplify knots using level moves. (A "level move" is like an under move, but there can be more strands lying below the arc that you move.) He says that he tried all unknots he could find on the web and they can all be fully monotonically simplified in very little time (the crossing number strictly decreases at each step, as far as I see).

For instance, the Gordian knot (shown below) can be fully monotonically simplified using level moves.

His program produces nice understandable pictures, and looking at them you can easily follow the moves that unknot the knot. These are available here

He actually wrote to us to ask for more examples to test the program with (there are not so many ready-to-use examples around), so if you know more hard unknots please share them (here or somewhere else)

This is just a comment. The same week (!) when Dylan asked this question, we received at our department a message from a non-professional mathematician who wrote a computer program that tries to simplify knots using level moves. (A "level move" is like an under move, but there can be more strands lying below the arc that you move.) He says that he tried all unknots he could find on the web and they can all be fully monotonically simplified in very little time (the crossing number strictly decreases at each step, as far as I see).

For instance, the Gordian knot (shown below) can be fully monotonically simplified using level moves.

His program produces nice understandable pictures, and looking at them you can easily follow the moves that unknot the knot. These are available here

He actually wrote to us to ask for more examples to test the program with (there are not so many ready-to-use examples around), so if you know more hard unknots please share them (here or somewhere else)

This is just a comment. The same week (!) when Dylan asked this question, we received at our department a message from a non-professional mathematician who wrote a computer program that tries to simplify knots using level moves. (A "level move" is like an under move, but there can be more strands lying below the arc that you move.) He says that he tried all unknots he could find on the web and they can all be fully monotonically simplified in very little time (the crossing number strictly decreases at each step, as far as I see). For instance, the Gordian knot can be fully monotonically simplified using level moves. His program produces nice understandable pictures, and looking at them you can easily follow the moves that unknot the knot. These are available here

He actually wrote to us to ask for more examples to test the program with, so if you know more hard unknots please share them (here or somewhere else)

This is just a comment. The same week (!) when Dylan asked this question, we received at our department a message from a non-professional mathematician who wrote a computer program that tries to simplify knots using level moves. (A "level move" is like an under move, but there can be more strands lying below the arc that you move.) He says that he tried all unknots he could find on the web and they can all be fully monotonically simplified in very little time (the crossing number strictly decreases at each step, as far as I see). 

For instance, the Gordian knot (shown below) can be fully monotonically simplified using level moves.

His program produces nice understandable pictures, and looking at them you can easily follow the moves that unknot the knot. These are available here

He actually wrote to us to ask for more examples to test the program with (there are not so many ready-to-use examples around), so if you know more hard unknots please share them (here or somewhere else)