Define the "diagram unknotting number" of a knot diagram $D$ as the minimal number of crossings that need to be changed in $D$ in order to get a diagram of the trivial knot (the usual unknotting number of a knot $K$ is the minimum over the diagram unknotting numbers of its diagrams).


Can you give me an example of a diagram of the trefoil knot (or any other knot having unknotting number = 1) with diagram unknotting number greater than 1?



Unknotting number of knot diagrams
It is a theorem of Stoimenow that there exist unknotting number one knots with minimal crossing diagrams of unknotting number greater than one. Two such examples are $14\_{36750}$ and $14\_{36760}$. See Figure 9 in the reference:


* A. Stoimenow. Some examples related to 4-genera, unknotting numbers and knot polynomials. J. London Math. Soc. (2), 63(2):487–500, 2001.


There is a related result of of Bleiler and Nakanishi (independently) that the knot $10\_8$ admits a 14-crossing diagram of unknotting number three --- yet all minimal crossing diagrams have unknotting number four! 


* Steven A. Bleiler. A note on unknotting number. Math. Proc. Cam-
bridge Philos. Soc., 96(3):469–471, 1984.
* Yasutaka Nakanishi. Unknotting numbers and knot diagrams with the
minimum crossings. Math. Sem. Notes Kobe Univ., 11(2):257–258, 1983.


There are is a discussion of these examples and some nice figures in [Staron's thesis](http://repositories.lib.utexas.edu/handle/2152/ETD-UT-2012-05-5055).


