Irregular smoothing and the number of Reidemeister moves
  In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking.   In this paper, we introduce a new link diagram invariant using irregular smoothing, and give an example of a knot diagram of the unknot for which the new invariant gives a better estimation than the old one. 
Chuichiro Hayashi and Miwa Hayashi
math.GT