Shadow movies not arising from knots
  A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister moves. This means that in Kaufman's theory of virtual knots, virtual crossings cannot simply be viewed as classical crossings where which strand is over has been left `to be determined'. 
Daniel Denton, Peter Doyle
math.GT