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Any knot diagram could be converted to an unknot by cross change. The unknotting number of a knot diagram is the minimal number of cross changes needed.

A knot could have many different diagrams and hence the corresponding unknotting numbers (of knot diagram). Pick up the smallest one and define it as the unknotting number of the knot.

There are cases where the unknotting number (of a knot) is achieved on a knot diagram without minimal crossing number. In general, it is a difficult problem to compute the unknotting number.

Question. Is there a good survey paper that collects the phenomena and typical tricks in this field?

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  • $\begingroup$ I believe there are examples where the unknotting arc isn't part of any minimal-crossing diagram. It's probably helpful to think of the unknotting arcs as embedded arcs in 3-space that connect the over/under crossing -- you are asking if these can be made to be vertical striaght-lines in a minimal planar diagram. Unfortunately I don't know these examples off-hand, but they typically appear in papers discussing unknotting number. i.e. from this perspective you can talk about unknotting arcs with respect to any diagram. $\endgroup$
    – Ryan Budney
    Commented Feb 19, 2021 at 1:38

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Lackenby discussed the unknotting number (and mentions some explicit knots for which we do not know the unknotting number!) in

Lackenby, Marc, Elementary knot theory, Woodhouse, N. M. J. (ed.), Lectures on geometry. Oxford: Oxford University Press (ISBN 978-0-19-878491-3/hbk). 29-64 (2017). ZBL1393.57002.

There are numerous references and a general discussion. In the arXiv version of this paper, the section on the unknotting number begins on page 23.

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