Knot Properties: Unknotting Number

The unknotting number of a knot is a property, denoted u(K) for a knot diagram K, that represents the fewest number of crossings that need to be changed to obtain the unknot. Again, the unknotting number is actually a property of the isotopic class and not necessarily of the projection symbolized by a knot diagram K. Like the bridge number, the unknotting number is not always recognized in the minimal diagram of the isotopic class.

The image below shows a trefoil knot with one crossing selected, followed by another image with the crossing "changed". It is quite easy to notice that a type II Reidemeister Move followed by a type I Reidemeister Move results in the unknot. Therefore the unknotting number of the trefoil knot is 1 because only one crossing needed to be changed.

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