  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.
