Can all knots be put as a series of over-under-over-under crossings? Sometimes knot diagrams show a segment of rope that crosses over two other segments, or a segment of rope that is crossed at either end but doesn't cross over anything. Are these knots equivalent to a knot in which all segments of rope cross over another segment exactly once? That is to say, is it possible to manipulate a knotted loop of rope such that the rope goes over itself, then under itself, then over, under, over, under, over, under, etc... without going over twice or under twice?

It is not always possible.
A knot where you can do this is called an alternating knot. The simplest non-alternating knots have 8 crossings; one of them is:

(image taken from the wikipedia article I linked to)
