Knot Diagram of the Trefoil with $4$ Crossings

Question

Let $D$ be a knot diagram with $4$ crossings. Suppose that $D$ represents the trefoil. Is it necessarily true that $D$ can be obtained from the standard diagram of the trefoil by applying the Reidermeister Move I once?

Answer 1

The following diagram is two moves away from a 3-crossing diagram of the trefoil.

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I found this by enumerating 4-valent graphs that could be knots (with some simplifying assumptions, there are only a couple), then assigning crossings, excluding figure-eights.