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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?

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    $\begingroup$ In a diagram, graph a strand of the knot and drag it inbetween all the crossings of the knot. I think that should be a good prescription. I'll see if I can draw a diagram. $\endgroup$ – Ryan Budney Oct 28 '14 at 20:20
  • $\begingroup$ Is it even clear that, for every knot $K$, there is some $n$ such that all diagrams of $K$ have unknotting number at most $n$? $\endgroup$ – Noah Schweber Oct 28 '14 at 21:22
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    $\begingroup$ I think for any knot you can construct diagrams with arbitrarily-high unknotting numbers. Presumably this is known and written down somewhere but I wouldn't know where to look. $\endgroup$ – Ryan Budney Oct 28 '14 at 21:29
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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.

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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.

example

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  • $\begingroup$ I think you can change a crossing and still get the unknot with one crossing change, although it's a bit hard to explain without using a picture myself. Take one of the strands that you looped around, and press the inner part of the strand under the strand of the "original" trefoil. Then the other two strands can pulled back, and there's an obvious reducing move. If you pushed the strands through each other (instead of just going over each other) then it might work. $\endgroup$ – Carl Nov 12 '14 at 22:23
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You should check this paper.

http://arxiv.org/abs/0805.3174

http://www.worldscientific.com/doi/abs/10.1142/S0218216509007361

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