Can all knots be put as a series of over-under-over-under crossings?

Question

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?

Answer 1

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)