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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)
Keywords:
2-bridge knot
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