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Unknotting number and knot diagram. (English) Zbl 0940.57008
The author proves the following theorem:
Let $$K$$ be a 2-bridge knot with unknotting number $$1$$. Then any minimum diagram of $$K$$ contains a crossing at which an application of unknotting operation yields a trivial knot.
This theorem need not be true for an arbitrary diagram of $$K$$, or if the unknotting number of $$K$$ is greater than one. In fact, the 2-bridge knot $$10_8$$ (whose unknotting number is $$2$$) gives a counterexample to the theorem. The proof of the theorem depends on the characterization of unknotting number one $$2$$-bridge knots due to Kanenobu and H. Murakami.
##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)
2-bridge knot
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