On link diagrams that are minimal with respect to Reidemeister moves I   and II
  In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof also shows that the number of crossings of such a minimal diagram is unique for any diagram of any link. As the unknot admits infinitely many non-trivial minimal diagrams, we see that every link has infinitely many minimal diagrams, by considering the connected sums with such diagrams. We show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II. 
Kishin Sasaki
math.GT