A knot can be represented with a Morse link presentation, as a combination of cups, caps and crossings (which is not uniquely determined by the knot, of course):
Two Morse link presentations of the same knot can be related by a sequence of the following moves:
- Swapping the order of two independent operations
- Pulling a cap, a cup or a crossing through two crossings (generalization of Reidemeister type 3)
- Cancelling out two successive crossings of alternating types (Reidemeister type 2)
- Twisting a cap or a cup (Reidemeister type 1)
- Introducing or eliminating a zig-zag
Given Morse presentation of the unknot, it seems to me that one can always simplify it to the trivial unknot diagram without ever introducing a zig-zag (so, never introducing new strands). So the last equality could be used in the simplifying direction only. Is this fact known and if so, how does the proof go? If not, I would be interested in a counter-example.