Monotonic simplification of ascending or descending knot diagrams

Establish that every ascending or descending knot diagram can be transformed, via a simplification (a sequence of Reidemeister moves that never increases the number of crossings), into the zero-crossing unknot diagram.

Background

The paper defines a simplification as a sequence of Reidemeister moves that never increases the number of crossings. While classical results show that some unknot diagrams ("hard unknots") cannot be monotonically simplified, the authors introduce a robot procedure that converts any knot diagram into an ascending diagram and then study simplification properties specific to ascending/descending diagrams.

They prove that loop detour moves can unknot ascending diagrams and further establish that, for ascending diagrams, the interior of loop-tangles can always be simplified so that the loop detour is itself a simplification, yielding a full simplification to a crossingless unknot diagram. They also show that the analogous statement fails for links. The conjecture asks for the complete monotonic simplification result for both ascending and descending knot diagrams.