Unknotting number of knot diagrams 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?
 A: You should check this paper.
http://arxiv.org/abs/0805.3174
http://www.worldscientific.com/doi/abs/10.1142/S0218216509007361
A: 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:


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*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! 


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*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.
A: Here is an example of what I'm thinking of.  You take a diagram of the trefoil, and grab strands of the knot, drag them around to crossings and slide them "between" the crossing.  I think all 1-step crossing changes are non-trivial knots for this example but I have not checked all cases. 

