Unknotting number

Remarks

The number $u(D)$ is finite: changing suitable crossings of any diagram gives a diagram of the unknot. Example: the standard three-crossing diagram of the trefoil knot becomes a diagram of the unknot after one crossing change. The trefoil is not the unknot, so its unknotting number is $1$. Non-example of the minimum: a single diagram may need more changes than $u(K)$, so $u(D)$ for one fixed diagram $D$ is only an upper bound for $u(K)$. A knot has $u(K) = 0$ if and only if it is the unknot. The unknotting number of the torus knot $T(p, q)$ with coprime $p, q \ge 2$ is $(p-1)(q-1)/2$; this is the Milnor conjecture.