Since the number of crossings in a knot diagram is the same as the number of arcs (or edges), can you construct a diagram of a knot where each arc crosses over exactly one other arc? My first thought would be to prove or disprove this with Reidermeister moves, but I couldn’t seem to find a way to make an argument because the 2nd and 3rd moves make things much more complicated.
Is there a way to prove or disprove this?